# Anisotropic Plasticity and Application to Plane Stress

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_225-1

## Synonyms

## Definitions

Plasticity is the ability of a material to deform in an irreversible manner. Unlike elasticity for which deformations are reversible, plasticity leads to permanent shape changes after a sufficient load is applied to a material. Anisotropic plasticity is the dependence of plastic properties on the loading direction. For plane stress states, loading is defined using only three stress components out of six for a general stress tensor. The description of plastic anisotropy for plane stress states is simpler than for general stress states but it has a wide range of practical applications in processes such as sheet forming.

## Introduction

In uniaxial tension, the stress-strain behavior of metals and alloys at low strain is, at first order, reversible and linear. This behavior is well described by Hooke’s law, in which the stress is proportional to the strain through the elastic modulus *E*. This law can be generalized to all the stress and strain components of the respective tensors. For an isotropic material, this introduces Poisson’s ratio *ν*, which linearly relates the transverse to longitudinal strains. When *E* and *ν* depend on the testing direction, the material is elastically anisotropic.

The elastic range, however, is bounded in uniaxial tension by the yield stress *σ* _{ y } above which plastic deformation occurs. For a biaxial stress state, yielding is described by a plane curve in stress space, the yield locus, which separates elastic from elastoplastic states. This curve is often represented as the von Mises ellipse for an isotropic material. The yielding concept can be generalized for a higher dimension, and the limit between elastic and elastoplastic stress states is generally called the yield surface. If the yield stress depends on the testing direction, the material is said to be plastically anisotropic. In general, it is considered that plastic deformation occurs without volume change and that hydrostatic pressure has virtually no influence on yielding. This is a consequence of the microscale deformation mechanisms that occur during plastic deformation. A number of additional phenomena have been observed when a material deforms plastically, but they are neglected in this entry.

Finite element (FE) simulations of sheet metal forming processes are very useful to optimize processes, and thereby, decrease development time and cost. Accurate results are achievable if sufficient consideration is given to the choice of the numerical parameters, including type of mesh, boundary conditions, and material constitutive behavior. The latter and, in particular, the description of plastic anisotropy are the topic of this entry. The essential role of the yield surface concept for the constitutive description of the metal behavior is emphasized. The state of plane stress is considered in more details because it is often a good approximation in sheet forming process analyses.

## Tensors and Invariants

### Cauchy Stress

*dS*and unit normal

**n**is

*d*

**f**

**=**

**t**

*dS*where the components of

**t**, the traction vector, have the unit of a stress.

**t**can be decomposed on the three orthogonal unit vectors

**e**

_{ p }corresponding to a Cartesian reference frame, i.e.,

**t**

**=**

*t*

_{ p }

**e**

_{ p }, may be employed. The traction vector in any direction

**n**at a point in a solid can be expressed using the stress tensor, as explained below.

*, a second-order tensor. It can be represented in dyadic notation as*

**σ****e**

_{ p }corresponds to the three orthogonal unit vectors of a Cartesian reference frame. Because two indices are repeated, the right-hand side of Eq. (1) is the summation of nine terms.

*σ*

_{ pq }are the components of

*among which six are independent in view of the tensor symmetry.*

**σ****e**

_{ p }

**e**

_{ q }denotes the dyadic product of two vectors defined such that, operating on any vector

**n**, e.g.,

**n**⋅

**e**

_{ p }

**e**

_{ q }, it leads to another vector with a coefficient from the corresponding scalar product, i.e.,

*n*

_{ p }

**e**

_{ q }with

*n*

_{ p }=

**n**⋅

**e**

_{ p }.

A tensor of second order can also be represented in the more usual way of a 3 × 3 matrix \(\left [ \sigma _{pq} \right ]\) with row *p* and column *q*. The traction vector **t** operating on any infinitesimal facet of surface *dS* and unit normal **n** in a solid is given by **t** **=** **n** ⋅* σ*. The three principal values

*σ*

_{ k }are such that

**t**

**=**

**n**⋅

**σ=***σ*

_{ k }

**n**, that is, colinear to the surface normal. Nontrivial solutions are only obtained if \(\mathrm {det}\left ({\boldsymbol {\upsigma }}-\sigma _{k}{\mathbf {I}}\right )\) is equal to zero.

### Invariants

*δ*

_{ pq }denotes the Kronecker symbol, similar invariants \(\tilde {J}_{1} =0\), \(\tilde {J}_{2} =3\tilde {I}_{2}\left ( \tilde {\boldsymbol {\upsigma {}}}^{\prime } \right )\) and \(\tilde {J}_{3} =2\tilde {I}_{3}\left ( \tilde {\boldsymbol {\upsigma }}^{\prime } \right )\) can be introduced. Since \(\tilde {J}_{1} =0\) \(\tilde {\sigma {}}_{M}\) is added to \(\tilde {J}_{2}\) and \(\tilde {J}_{3}\) to form a system of three independent tensor invariants. It can be shown that \(\tilde {J}_{2}\) is always nonnegative. Using \(\tilde {I}_{3}\) or \(\tilde {J}_{3}\), the so-called angular invariant can be introduced, i.e.,

### Principal Values

*q*, the invariants in Eq. (3) simplify considerably and Eq. (2) becomes

### Principal Stresses

If \(\tilde {\boldsymbol {\upsigma {}}}\mathbf {=}{\boldsymbol {\upsigma {}}}\) is the stress tensor, the relationships in Eq. (5) with *I* _{1}, *I* _{2}, and *I* _{3} instead of \(\tilde {I}_{1}\), \(\tilde {I}_{2}\), and \(\tilde {I}_{3}\) lead to the principal stress values and those in Eq. (6) with *J* _{1} = 0, *J* _{2}, and *J* _{3}, to their deviatoric parts. For a state of plane stress, i.e., \(\tilde {\boldsymbol {\upsigma {}}}\mathbf {=}{\boldsymbol {\upsigma {}}}\) with *σ* _{3q } = *σ* _{ q3} = 0, the relationships in Eqs. (9) and (10) allow the calculation of the tensor and deviator principal stresses. The different expressions of the principal values of a tensor and its deviator are important for application to plastic anisotropy.

## Flow Theory of Plasticity

### Uniaxial Stress State

In this entry, the small elastic deformations are neglected as they can be added to the large plastic strains in an incremental manner. In uniaxial tension, a material becomes plastic when the applied stress *σ* reaches a critical value, the yield stress *σ* _{ y }. Thus, *σ* − *σ* _{ y } = 0 is called the yield condition. However, this critical stress increases as plastic deformation accumulates through the phenomenon of strain hardening. Thus, the flow stress associated to a tensile stress-strain curve \(\sigma _{f}\left ( \varepsilon \right )\) can replace *σ* _{ y } in the yield condition. For an isotropic material, the transverse strains are equal to minus half of the longitudinal strain. This determines the flow rule, i.e., the components of the plastic strain increment. The flow theory is an incremental approach because plasticity is a highly nonlinear phenomenon. Yield condition, hardening, and flow rule constitute the foundation of the flow theory of plasticity.

### Multiaxial Stress State

*σ*

_{ R }is a reference flow curve not necessarily obtained from the uniaxial tension test, and \(\bar {\varepsilon {}}\) is the effective strain defined from the plastic work equivalence condition, i.e., \(\bar {\sigma {}}d\bar {\varepsilon {}}=\sigma _{pq}d\varepsilon _{pq}\) where the components of the plastic strain increment tensor,

*dε*

_{ pq }, are defined below. In this entry, isotropic hardening is assumed, which indicates that the normalized yield condition, Open image in new window , remains identical during plastic deformation. The associated flow rule is generally suitable for metals, that is, the strain increment is proportional to the gradient of the yield function as a consequence of the microscopic deformation mechanisms that occur during plastic deformation

### Isotropic Yield Conditions

*σ*

_{ S }. The corresponding yield surface is a cylinder of axis defined by

*σ*

_{1}=

*σ*

_{2}=

*σ*

_{3}in principal stress space, with a hexagonal cross-section. The most popular yield condition, proposed by von Mises, is quadratic and expressed in an expanded form as

*σ*

_{ R }denotes the flow stress in uniaxial tension. The von Mises effective stress can be expressed using either the principal stress or deviatoric components. The corresponding yield surface is a cylinder of axis defined by

*σ*

_{1}=

*σ*

_{2}=

*σ*

_{3}in principal stress space, with a circular cross-section. Note that, using the principal deviatoric stresses expressed in Eq. (6), the yield condition in (12) reduces to the compact form \(\sqrt {3J_{2}} =\sigma _{R}\), which explains why it is called the

*J*

_{2}-flow theory of plasticity. This approach was further extended by Drucker (1949) to \(3J_{2}^{3}-cJ_{3}^{2}=\sigma _{S}^{6}\) where

*c*is a coefficient. Another description of a cylindrical yield surface, which includes Tresca and von Mises as particular cases, was proposed by Hershey (1954) to approximate yield surfaces calculated with a crystal plasticity model For

*a*= 2 or

*a*= 4, Eq. (13) reduces to the Mises yield function, whereas for

*a*= 1 and in the limiting case

*a*→

*∞*, it leads to Tresca. According to Logan and Hosford (1980), a particularly good agreement with many crystal plasticity results can be obtained using Eq. (13) with

*a*= 6 and

*a*= 8 for BCC and FCC materials, respectively. Karafillis and Boyce (1993) proposed a generalization of Hershey’s model where

*c*is a coefficient between 0 and 1. Cazacu and Barlat (2005) introduced an isotropic yield function that accounts for the strength differential (SD) effect in pressure-independent materials such as hexagonal (HCP) polycrystals where

*K*is a coefficient. The literature on yield conditions is wide as, for instance, Życzkowski (1981) and Yu (2002) listed several hundred. A general stress state reduces to plane stress in Eqs. (12), (13), (14), and (15) when one of the principal stresses

*σ*

_{ k }is equal to zero.

## Anisotropic Plasticity

### Material Axes

There are a number of ways to describe plastic anisotropy (Skrzypek and Ganczarski, 2015). This entry mainly discusses the formulation introduced by Hill (1948) and the framework based on linear transformation of the stress tensor. Anisotropy is the dependence of material properties on the testing direction. Unlike isotropic theories for which the choice of a reference frame is arbitrary, anisotropic formulations must be expressed in a system of axes attached to the material. For instance, due to prior thermomechanical processing history, sheet metals exhibit orthotropic symmetry, i.e., they possess three mutually orthogonal planes of symmetry. In this paper, the unit vectors associated to these planes are denoted **e** _{ x }, **e** _{ y }, and **e** _{ z }, which correspond to the sheet rolling (RD), transverse (TD) and normal (ND) directions, respectively. In the yield condition, the stress invariants should be replaced by common invariants of the stress and material symmetry tensors.

### Quadratic Formulations

*σ*

_{ M }, contains 15 anisotropy coefficients. As a particular case, Hill (1948) proposed the following quadratic form for orthotropic materials with six coefficients,

*F*,

*G*,

*H*,

*L*,

*M*, and

*N*

### Linear Tensor Transformations

*” denotes the double dot product, i.e.,*

**:****L**

**:***=*

**σ***L*

_{ pqrs }

*σ*

_{ rs }

**e**

_{ p }

**e**

_{ q }.

**T**transforms the Cauchy stress tensor into its deviator, i.e.,

**σ**^{ ′ }=

**T**

**:***. Each of the fourth-order tensors*

**σ****C**and

**L**, expressed in the material symmetry axes, contains a set of anisotropy coefficients. The structure of such tensors is similar to that of the elastic moduli for crystals or polycrystals with the suitable symmetry, although the condition

*C*

_{ pqrs }=

*C*

_{ rspq }is not necessary for plastic anisotropy. For instance, for orthotropic symmetry such as in rolled sheets and plates, a matrix representation of \(\tilde {\boldsymbol {\upsigma }}^{\prime }={\mathbf {C}\boldsymbol {:\sigma }}\) with second-order tensors as columns with six rows and a fourth-order tensor as a 6×6 matrices can be written as follows:

*C*

_{ pq }, which are different from but related to

*C*

_{ pqrs }, are not independent.

### Non-quadratic Approaches

**C**

**:****T**

**=**

**L**was proposed by Barlat et al. (1991) as a yield condition called Yld91. Note that the reference stress is not necessarily the uniaxial stress. Moreover, if

**C**is the fourth-order identity tensor, Eq. (21) reduces to the isotropic function in Eq. (13). In a similar way, a more general isotropic yield function in Eq. (14) and different types of material symmetries were considered in Karafillis and Boyce (1993). The isotropic approach that accounts for a SD effect in Eq. (15) was extended to orthotropic anisotropy in Cazacu and Barlat (2006).

*t*of linear transformations similar to Eq. (19)

*t*= 1 and 2, Bron and Besson (2005) extended Eq. (14) for orthotropic anisotropy, and Barlat et al. (2007) proposed the so-called Yld2004-18p which generalizes Eq. (21). The number of terms in Eq. (23) is nine. This is an isotropic function because the indices

*p*and

*q*permute. Although 18 coefficients are available in this yield condition, van den Boogaard et al. (2016) demonstrated that only 16 are independent. Note that the transformed stresses are not necessarily deviatoric, i.e., \(\tilde {I}_{1}\) is possibly not 0, even though the formulation is independent of the mean stress. Thus, the principal stresses

*σ*

_{1},

*σ*

_{2}, and

*σ*

_{3}in Eq. (5) must be used to calculate the principal values of the transformed stress tensor \({{\boldsymbol {\upsigma {}}}}^{\prime {\left ( t \right )}}\). Aretz and Barlat (2013) proposed different isotropic forms for extension to anisotropy. Based on the same principle, Cazacu et al. (2006) extended Eq. (15) further with two linear transformations.

Non-quadratic formulations for plastic anisotropy, such as Eqs. (21) or (23) are applicable for plane stress, say, in direction **e** _{3}, by simply assuming *σ* _{3q } = *σ* _{ q3} = 0 in Eqs. (19) and (22). The principal values of the transformed stress tensor \(\tilde {\boldsymbol {\upsigma {}}}\) or \({\boldsymbol {\upsigma {}}}^{\left ( \mathrm{t} \right )}\) are obtained from Eqs. (5).

## Specific Plane Stress Approach

### Yield Function

*ϕ*and

*ψ*are two isotropic functions since it is possible to permute the (in-plane) indices 1 and 2 in each function. Using two linear transformations as in Eq. (22), the isotropic yield condition in Eq. (24) can be generalized to plastic anisotropy, i.e.,

### Anisotropy Coefficients

### Anisotropy Coefficient Tensors

*α*

_{ j }and \(C_{mn}^{\left ( t \right )}\) in Eqs. (33) and (36), the matrix representations of tensors \({\mathbf {C}}^{\left ( t \right )}\) become

*α*

_{ j }(for

*j*= 1 to 8) reduce to 1 in the isotropic case. Using Eq. (22), the matrix representations of tensors \({\mathbf {L}}^{\left ( t \right )}\) can be expressed as follows:

*a*, eight coefficients

*α*

_{ q }are necessary to completely define the yield condition. If

*a*is assumed to be six or eight, as recommended for BCC and FCC materials, respectively, a numerical solver, such as Newton-Raphson, can be employed to calculate the coefficients

*α*

_{ q }from a set of eight experimental data. However, the identification can be optimized, including for the value of the exponent

*a*using other or additional input data, which is discussed in the next section.

### Flow Rule

*dε*

_{ pq }that can be solved using limit calculations (Barlat et al., 2003a).

## Identification and Validation

So far, a general method suitable for the development of mathematical descriptions of plastic anisotropy was discussed. A specific formulation was detailed for plane stress, which is a reasonable approximation of a stress state in sheet metal forming processes. However, in order to determine the anisotropy coefficients for a given material, it is necessary to generate data from multiaxial stress states to characterize the specific anisotropy of a sheet metal. The remaining of this entry focuses on mechanical testing techniques to acquire suitable data and on approaches needed to extract the anisotropy coefficients.

## Conventional Mechanical Tests

### Uniaxial Tension Test

Uniaxial tensile tests are performed to determine the basic mechanical properties of sheet metals, such as a stress-strain curve, yield stress, uniform elongation, and total elongation, as specified in ISO (2009). ISO (2006) specifies a method for determining the r-value, one of the important parameters for characterizing anisotropy of sheet metals. ISO (2007) specifies a method for determining the tensile strain hardening exponent *n* of metallic flat products (sheet and strip).

### Plane Strain Tension Test

*W*, and the difference in the tensile force

*ΔF*is measured. Thus, the flow stress under plane strain tension,

*σ*

_{ ps }, is calculated as

*t*

_{0}is the initial sheet thickness and

*ΔW*is the difference in the width of the gauge section. It is reported that the specimen geometry should satisfy

*W*/(2

*R*) ≥ 12 so that plane strain tension is maintained in the central part of the gauge section (An et al., 2004).

**y−**direction in Fig. 2a, is invariably found to differ from the “Force/Area” estimation. This comparison then provides a correction coefficient for the “Force/Area,” which yields the stress at the center of the specimen.

A shortcoming of this approach, but one that is perhaps inherent to the plane strain tension specimen geometries, is that it is not possible to determine the transverse stress inside the gauge section, i.e., **x** −direction in Fig. 2a. However, since the von Mises yield criterion has already been assumed, it can be used to determine the transverse stress. These two stresses are then used to calibrate an anisotropic yield criterion. An alternative for the calibration of the yield criterion is to use the stress along the loading direction and the condition of plane strain deformation. In either case, the simulation is rerun with this criterion and the process is repeated, further refining the determination of the stress state during plane strain deformation.

### Hydraulic Bulge Test

*ρ*as well as the thickness strain at the apex of the bulge specimen are measured to obtain an equibiaxial tensile stress

*σ*

_{ b }– logarithmic thickness strain \(\varepsilon _{3}^{p}\) curve.

*σ*

_{ b }is calculated as

*σ*

_{ b }=

*Pρ*/(2

*t*) in which the evaluation of

*ρ*at the middle plane of the sheet is recommended. The plastic thickness strain, \(\varepsilon _{3}^{p}=-\varepsilon _{1}^{p}-\varepsilon _{2}^{p}\), is calculated from the condition of volume conservation, where \(\varepsilon _{1}^{p}\) and \(\varepsilon _{2}^{p}\) are the in-plane normal plastic strain components at the apex of the specimen.

ISO (2014a) specifies a method for the determination of the \(\sigma _{b}-\varepsilon _{3}^{p}\) curve of metallic sheets having a thickness below 3 mm. Mulder et al. (2015) discussed the effects of several parameters in the test on the accuracy of a \(\sigma _{b}-\varepsilon _{3}^{p}\) curve measured following ISO (2014a). Yoshida (2013) performed finite element analyses of the hydraulic bulge test to conclude that the measurement error of *σ* _{ b } is less than 1% for an isotropic material when Open image in new window , Open image in new window , Open image in new window , where *D* _{d} is the diameter of the die hole, *D* _{ ρ } is the initial gauge length for the measurement of *ρ*, *D* _{ ε } is the initial gauge length for the measurement of \(\varepsilon _{1}^{p}\) and \(\varepsilon _{2}^{p}\), and *a* is the initial sheet thickness. For an accurate determination of biaxial stress-strain curves, the strain rate must be constant during bulging (Ranta-Eskola, 1979).

## Advanced Mechanical Tests

This section describes the material testing methods for measuring and quantifying the deformation behavior of sheet materials under biaxial stress states. The rolling (RD), transverse (TD), and thickness (ND) directions of an original flat sheet sample are defined as the **x** −, **y** −, and **z** −axes, respectively, throughout this section.

### Biaxial Compressive Test Using an Adhesively Bonded Sheet Laminate Specimen

*π*−plane (Tozawa, 1978). It is noted that when the plastic deformation mechanism of the material is influenced by the hydrostatic component of stress (Spitzig and Richmond, 1984), the yield locus shapes obtained from the biaxial compression test may differ from those obtained from the biaxial tension test (Lowden and Hutchinson, 1975).

### Biaxial Tensile Test Using a Cruciform Specimen

ISO (2014b) specifies the testing method for measuring the biaxial stress-strain curves of sheet metals subject to biaxial tension at an arbitrary stress ratio using a cruciform specimen from a flat sheet metal. The standard applies to the shape and strain measurement position for the cruciform test piece. Other types of cruciform specimens are reviewed by Kuwabara (2007) and Hannon and Tiernan (2008).

- (a)
The specimen shall be cut from as-received flat-rolled sheet samples so that the arms are parallel to the RD and TD of the sheet sample. For the fabrication of the specimen (including machining of slits), laser or water-jet machining is recommended.

- (b)
Seven slits per one arm shall be made. The slit length

*L*should be in the range*B*≤*L*≤ 2*B*. The slit width*w*_{S}should be made as small as possible, and should be less than 0.01*B*. - (c)
When the material thickness,

*t*_{0}, exceeds 8% of the arm width*B*, either the arm width*B*shall be increased in such a manner that satisfies*t*_{0}≤ 0.08*B*or the specimen thickness shall be reduced.

According to the FE analysis of the specimen geometry as shown in Fig. 5, the error of stress measurement is estimated to be less than 2% (Hanabusa et al., 2010, 2013) when the strain measurement position is chosen as shown in Fig. 5b. The maximum equivalent plastic strain \(\varepsilon _{max}^{p}\) applicable to the gauge area depends mainly on the stress ratio, the work hardening exponent (*n*-value) of the test material, the slit width, and the anisotropy of the material (Hanabusa et al., 2013). \(\varepsilon _{max}^{p}\) can be increased by reducing the thickness of the gauge section of a cruciform specimen (Deng et al., 2015).

### Combined Biaxial Tension-Compression Test

### In-Plane Uniaxial Stress Reversal Test

In bending of a sheet material, a part of the material fibers layered in the thickness direction of the sheet are subjected to in-plane compression. The flow stress of some metals is often observed to be higher in compression than in tension, and vice versa, even if the metal is fully annealed; this phenomenon is referred to as the strength-differential (SD) effect (Spitzig et al., 1975). Moreover, the directionality of dislocation structures in severely deformed metals and their rearrangement during reloading can result in tension-compression asymmetry (TCA) of the flow stress in the material (Barlat et al., 2003b; Yapici et al., 2007). Consequently, in order to perform accurate forming simulations for sheet metals, it is vital to develop a testing device that enables correct measurement of the stress-strain responses of the sample subjected to in-plane compression and/or in-plane stress reversal and to establish more sophisticated material models that enable the reproduction of the observed SD effect and/or TCA.

### Multiaxial Tube Expansion Test

The maximum plastic strain \(\varepsilon _{max}^{p}\) applicable to the cruciform specimen shown in Fig. 5 is less than 0.05 in most ductile sheet metals. The specimen proposed by Deng et al., 2015 can achieve larger strains, but it can be cumbersome to manufacture. The multiaxial tube expansion testing method is a useful material testing method to measure the biaxial deformation behavior of sheet metals for a large plastic strain range. This section introduces the multiaxial tube expansion testing method. The reader is referred to the excellent reviews of earlier work on multiaxial tube expansion/torsion testing methods (Hecker, 1976; Michno and Findley, 1976; Ikegami, 1979; Phillips, 1986), although most of the studies were concerned with the initial yielding behavior of tubular specimens machined from bulk metals. An application of the method to larger strains was presented by Hill et al., 1994.

*ε*

_{ ϕ }and

*ε*

_{ θ }, are evaluated at midthickness using the following equations:

*ε*

_{ ϕS}and

*ε*

_{ θS}are the axial and circumferential logarithmic strain components measured on the outer surface of the specimen,

*D*

_{0}and

*t*

_{0}are the initial outer diameter and wall thickness of the specimen, and

*D*,

*t*, and

*R*

_{ ϕ }are the outer diameter, thickness, and axial radius of curvature measured at the center of the bulging specimen. The current thickness

*t*is determined under the assumption of constant volume and neglecting elastic strains as

*σ*

_{ ϕ }and

*σ*

_{ θ }, are calculated as the values at midthickness, using the following equations based on the equilibrium requirements at the center of the specimen:

*T*and

*P*are measured values of the axial load and internal pressure, respectively.

The measured values of *T*, *P*, *ε* _{ ϕ }, *ε* _{ θ }, and *R* _{ ϕ } are input to a personal computer to calculate *σ* _{ ϕ } and *σ* _{ θ } using Eqs. (44) and (45). The calculated values are then compared with the command values from the controller to impose the axial force and internal pressure so that the predetermined stress path (*σ* _{ ϕ }, *σ* _{ θ }) can be applied to the specimen.

### Combined Tension-Shear Test

Multiaxial stress testing methods with the application of normal and tangential loads to a rectangular sheet specimen have been proposed by Vegter and van den Boogaard (2006) and Mohr and Oswald (2008). For tubular materials, and beyond the tension-torsion experiment, which requires specialized testing equipment, Dick and Korkolis (2015) proposed a method to apply combined tension and torsion using a universal testing machine.

## Applications

### Application to Steel

*α*

_{ i }(

*i*= 1 − 8) and exponent

*a*of the Yld2000-2d yield function are determined to minimize the cost function given by

*N*is the number of stress points forming the work contour,

*l*\(=\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{xy}^{2}} \) (the distance between the origin of the

*σ*

_{ x }−

*σ*

_{ y }−

*σ*

_{ xy }stress space and the

*j*

^{th}stress point forming a contour of equal plastic work),

*l*

^{ ′ }the distance between the origin of the

*σ*

_{ x }−

*σ*

_{ y }−

*σ*

_{ xy }stress space and the calculated yield locus along the stress path to the

*j*

^{th}stress point,

*β*the direction of the plastic strain rate measured at the

*j*

^{th}stress path, and

*β*

^{ ′ }the direction of the plastic strain rate calculated using the Yld2000-2d yield function and the associated flow rule. A real-coded genetic algorithm was used to minimize the cost function and to avoid a local optimum solution.

*F*. In the method A, only uniaxial tension test data are used, while, in the method B, the equibiaxial tension test data, as well as the uniaxial tension test data, are taken into account. The uniaxial tension test data used are those measured in the directions of

*θ*= 0, 15, 30, 45, 60, 75, and 90

^{∘}from the rolling direction (RD). Table 1 shows the values of

*w*

_{ j,l }and

*w*

_{ j,β }used in the methods A and B. The value of

*w*

_{1,l }(

*θ*= 0) was chosen to be 1000 so that the equivalent stress calculated using the Yld2000-2d yield function should be consistent with the uniaxial tensile flow stress for the RD.

Values of *w* _{ j,l } and *w* _{ j,β } used for the methods A and B

Method A | Method B | |||
---|---|---|---|---|

| 7 | 8 | ||

Tensile direction from RD | | | | |

1000 | 0.2 | 1000 | 0.2 | |

15, 30, 45, 60, 75, 90 | 1 | 0.2 | 1 | 0.2 |

Equibiaxial | NA | NA | 1 | 0.2 |

*α*

_{ i }(

*i*= 1 − 8) and

*a*based on the methods A and B are shown in Table 2. In the method A, three different initial values of

*a*, namely, 6, 10, and 20, referred to as A-1, 2, and 3, respectively, were assumed. The reference experimental data for the work contours and the directions of the plastic strain rate were those measured at the reference plastic strain of \(\varepsilon _{0}^{p}=0.09\), where \(\varepsilon _{0}^{p}\) is associated with the plastic work per unit volume dissipated in the uniaxial tensile test in the RD up to a logarithmic tensile plastic strain of \(\varepsilon _{0}^{p}\).

Calculated values of *α* _{ i } (*i* = 1 − 8) and *a* based on the methods A and B

Initial | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

A-1 | 6 | 5.7037 | 1.0063 | 1.1248 | 1.1975 | 0.9429 | 0.9706 | 1.0582 | 1.0660 | 0.8232 |

A-2 | 10 | 8.0130 | 1.0290 | 1.0690 | 1.0016 | 0.9405 | 1.0149 | 1.2414 | 1.0440 | 0.8488 |

A-3 | 20 | 16.1297 | 1.0181 | 1.0297 | 0.8805 | 0.9432 | 1.0439 | 1.2749 | 1.0241 | 0.9257 |

B | 6 | 6.8071 | 0.9887 | 1.1019 | 0.8990 | 0.8988 | 0.9216 | 0.7799 | 1.0412 | 1.0321 |

*a*. Figure 9b compares the directions of the plastic strain rates measured at different levels of \(\varepsilon _{0}^{p}\) with those calculated using the Yld2000-2d yield function. Again, the predictions by the method

*B*are consistent with the measurements.

### Application to Aluminum Alloy

In this section, an example of material modeling using the Yld2000-2d yield function is presented for an aluminum alloy, namely, a 1 mm thick 6022-T4 sheet sample, which is a typical material for automotive body applications. To provide a different perspective from the previous section, the anisotropy of this alloy is calibrated using only a universal testing machine. For this, uniaxial tensile (UT) specimens at every 15^{∘} to the RD, plane strain tension (PST) specimens at 45^{∘} to the RD, as well as disc specimens for through-thickness compression (DC) are prepared. The off-axis UT and PST tests provide loading under combined tension and shear, which enables a more complete calibration of the yield function. Under conditions of plastic incompressibility, the DC test is equivalent to equibiaxial tension in the plane of the sheet. However, due to the nature of this test and in particular, the inability to completely negate the effects of friction and tool compliance on the results, the DC test is considered effective only for providing the plastic flow ratio under equibiaxial tension, rather than the stresses themselves.

^{3}(which corresponds to a strain of 2.5% in uniaxial tension).

The plane strain tension tests are used to provide three additional inputs for calibrating the anisotropic yield function. As described earlier (section “Plane Strain Tension Test”), the stress and strain fields are not uniform inside the gauge section of the PST specimen, with plane strain deformation prevailing at the center and uniaxial tension as the edges are approached. Therefore, the stress in the loading direction cannot be estimated simply as “Force/Area.” This complication was overcome with the aid of DIC and finite element analysis, as described in the earlier section. The correction coefficient for this specimen geometry and material saturates around 0.96 in the moderate and large post-yield strain region.

The DC tests provide the plastic flow ratio under equibiaxial tension. This is done by periodically interrupting the test and measuring the dimensions of the deformed specimen along the RD and TD. Special biaxial extensometers have also been proposed in the literature, though they are oftentimes used for the stack compression test, which allows for a distance between the compression platens that is a multiple of the initial sheet thickness (Coppieters et al., 2010). Since the test is not limited by the localization of deformation typical of the UT and PST specimens, significantly larger strains can be attained.

*a*is taken to be 8, as suggested by Logan and Hosford (1980) for FCC metals. The eight anisotropy coefficients

*α*

_{1}to

*α*

_{8}are calibrated by fitting the experiments with the least squares method. To overcome the uncertainty of determining the transverse stress in the PST tests, the plane strain condition for the RD and TD tests was adopted instead. The resulting parameters are listed in Table 3. The agreement of the Yld2000-2d criterion with the experimental yield locus is shown in Fig. 12a and with the plastic strain ratios in Fig. 12b. In both spaces, the agreement of the model to the experiments is very good, justifying the effort of determining the eight anisotropy parameters.

Calculated values of *α* _{ i } (*i* = 1 − 8) and *a* for AA6022-T4

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

8 | 0.919 | 1.072 | 1.204 | 1.109 | 0.992 | 0.892 | 0.957 | 1.098 |

## Concluding Remarks

This entry discussed theoretical and experimental approaches for the description of plastic anisotropy and the complementarity of these methods. The plane stress case, with only three non-zero stress components from a total of six in the general case, is reviewed in more details because it is simpler to treat theoretically and experimentally but, still, can be applied in numerous meaningful applications such as sheet metal forming simulations.

For metals, the description of plastic anisotropy reduces essentially to the definition of a yield condition, which can be viewed as a surface in plane stress space. The yield condition must obey a number of constraints in order to be consistent with the laws of physics and mechanics. The experiments have two main purposes, namely, provide data to determine anisotropy coefficients and validate the adequacy of the formulation with the observed behavior.

The examples provided highlight the fact that plastic anisotropy can be quite complex even under plane stress conditions. This justifies a posteriori the development of advanced anisotropic yield conditions, as described in sections “Anisotropic Plasticity” and “Specific Plane Stress Approach”. However, these examples also underscore the fact that the calibration of the yield functions should be treated carefully, to ensure that the real material deformation behavior is accurately captured. Only under this condition, the application of anisotropic plasticity to numerical forming simulations can be successfully exploited for the optimization of real processes.

## Cross-References

## Notes

### Acknowledgements

The authors gratefully acknowledge the supports of POSCO and the Global Innovation Research Organization in TUAT for this entry. In addition, the authors are indebted to Dr. Tomoyuki Hakoyama (The RIKEN) for help with the parameter identification analysis in section “Application to Steel”.

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