Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Anisotropic Plasticity and Application to Plane Stress

  • Frédéric BarlatEmail author
  • Toshihiko Kuwabara
  • Yannis P. Korkolis
Living reference work entry
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

In a solid, the force acting on any infinitesimal facet of surface dS and unit normal n is df = tdS 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.,
$$\displaystyle \begin{aligned} {\mathbf{t=}}t_{1}{\mathbf{e}}_{1}+t_{2}{\mathbf{e}}_{2}+t_{3}{\mathbf{e}}_{3}=t_{p}{\mathbf{e}}_{p}\end{aligned} $$
in which the summation convention on repeated indices, 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.
The main variable selected to describe plasticity is assumed to be the Cauchy stress σ, a second-order tensor. It can be represented in dyadic notation as
$$\displaystyle \begin{aligned} {\boldsymbol{\upsigma}} =\sigma_{pq}{\mathbf{e}}_{p}{\mathbf{e}}_{q}\ (\text{also}\\text{denoted}\ \sigma_{pq}{\mathbf{e}}_{p}\otimes{\mathbf{e}}_{q}) \end{aligned} $$
(1)
where 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 σ among which six are independent in view of the tensor symmetry. 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

Instead of six independent components, a general symmetric tensor of second-order \(\tilde {\boldsymbol {\upsigma {}}}\), possibly the Cauchy stress, can be represented with a set of three invariants. Although a number of sets can be defined (Życzkowski, 1981), the principal invariants of \(\tilde {\boldsymbol {\upsigma }}\) are obtained by solving the characteristic cubic equation that provides the three principal stresses \(\tilde {\sigma {}}_{k}\)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \tilde{\sigma{}}_{k} \right)&=&\mathrm{det}\left( \boldsymbol{\tilde{\upsigma }}\mathbf{-}\tilde{\sigma{}}_{k}{\mathbf{I}} \right)=-\tilde{\sigma }_{k}^{3}+3\tilde{I}_{1}\tilde{\sigma{}}_{k}^{2}\\&+&3\tilde{I}_{2}\tilde{\sigma }_{k}+2\tilde{I}_{3}=0 \end{array} \end{aligned} $$
(2)
The coefficients of Eq. (2) are called the principal invariants
$$\displaystyle\begin{array}{rcl} \tilde{I}_{1}& =& \left (\tilde{\sigma }_{11} +\tilde{\sigma } _{22} +\tilde{\sigma } _{33}\right )/3 =\tilde{\sigma } _{M} \\ \tilde{I}_{2}& =& \left (\tilde{\sigma }_{23}^{2} +\tilde{\sigma }_{ 31}^{2} +\tilde{\sigma }_{ 12}^{2} -\tilde{\sigma }_{ 22}\tilde{\sigma }_{33} -\tilde{\sigma }_{33}\tilde{\sigma }_{11} -\tilde{\sigma }_{11}\tilde{\sigma }_{22}\right )/3 \\ \tilde{I}_{3}& =& \left (2\tilde{\sigma }_{23}\tilde{\sigma }_{31}\tilde{\sigma }_{12} +\tilde{\sigma } _{11}\tilde{\sigma }_{22}\tilde{\sigma }_{33} -\tilde{\sigma }_{11}\tilde{\sigma }_{23}^{2} -\tilde{\sigma }_{ 22}\tilde{\sigma }_{31}^{2} -\tilde{\sigma }_{ 33}\tilde{\sigma }_{12}^{2}\right )/2{}\end{array}$$
(3)
\(\tilde {I}_{1}=\tilde {\sigma {}}_{M}\) is also called the mean value of \(\tilde {\boldsymbol {\upsigma {}}}\). Moreover, from the deviator \(\tilde {\sigma }_{pq}^{\prime }=\tilde {\sigma {}}_{pq}-\tilde {\sigma {}}_{M}\delta _{pq}\), where 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.,
$$\displaystyle{ \tilde{\theta }=\mathrm{ arcos}\left \{\frac{2\tilde{I}_{1}^{3} + 3\tilde{I}_{1}\tilde{I}_{2} + 2\tilde{I}_{3}} {2\left (\tilde{I}_{1}^{2} +\tilde{ I}_{2}\right )^{-3/2}} \right \} }$$
(4)

Principal Values

From the different invariants defined above, the ordered principal values of \(\boldsymbol {\tilde {\upsigma {}}}\), \(\tilde {\sigma {}}_{1}\ge \tilde {\sigma {}}_{2}\ge \tilde {\sigma }_{3}\) can be calculated (see Barlat et al., 1991, 2005)
$$\displaystyle\begin{array}{rcl} \begin{array}{l} \tilde{\sigma }_{1} = 2\sqrt{\tilde{I}_{1 }^{2 } +\tilde{ I}_{2}}\mathrm{cos}\left \{\tilde{\theta }/3\right \} +\tilde{ I}_{1} \\ \tilde{\sigma }_{2} = 2\sqrt{\tilde{I}_{1 }^{2 } +\tilde{ I}_{2}}\mathrm{cos}\left \{\left (\tilde{\theta }+4\pi \right )/3\right \} +\tilde{ I}_{1} \\ \tilde{\sigma }_{3} = 2\sqrt{\tilde{I}_{1 }^{2 } +\tilde{ I}_{2}}\mathrm{cos}\left \{\left (\tilde{\theta }+2\pi \right )/3\right \} +\tilde{ I}_{1}\end{array} & &{}\end{array}$$
(5)
If \(\tilde {I}_{1}=0,\) the corresponding deviatoric values of \(\tilde {\sigma }\) reduce to Alternatively, if \(\tilde {\sigma {}}_{3q}=\tilde {\sigma {}}_{q3}=0\) for any value of q, the invariants in Eq. (3) simplify considerably and Eq. (2) becomes
$$\displaystyle \begin{aligned} P\left( \tilde{\sigma{}}_{k} \right)=-\tilde{\sigma }_{k}^{2}+3\tilde{I}_{1}\tilde{\sigma{}}_{k}+2\tilde{I}_{2}=0 \end{aligned} $$
(8)
The expression of the principal values for the tensor and its deviator becomes simpler
$$\displaystyle \begin{aligned} \begin{array}{l} \tilde{\sigma{}}_{1}= \frac{\tilde{\sigma }_{11}+\tilde{\sigma{}}_{22}}{2}+\sqrt{\frac{\left( \tilde{\sigma{}}_{11}-\tilde{\sigma }_{22} \right)^{2}}{2}-\tilde{\sigma{}}_{12}^{2}} \\ {} \tilde{\sigma{}}_{2}= \frac{\tilde{\sigma }_{11}+\tilde{\sigma{}}_{22}}{2}-\sqrt{\frac{\left( \tilde{\sigma{}}_{11}-\tilde{\sigma }_{22} \right)^{2}}{2}-\tilde{\sigma{}}_{12}^{2}} \\ \tilde{\sigma}_{3}=0 \end{array} \end{aligned} $$
(9)
and
$$\displaystyle \begin{aligned} \begin{array}{l} \tilde{\sigma{}}_{1}^{\prime}=\frac{\tilde{\sigma }_{11}+\tilde{\sigma{}}_{22}}{6}+\sqrt{\frac{\left( \tilde{\sigma{}}_{11}-\tilde{\sigma }_{22} \right)^{2}}{2}-\tilde{\sigma{}}_{12}^{2}} \\ {} \tilde{\sigma{}}_{2}^{\prime}=\frac{\tilde{\sigma }_{11}+\tilde{\sigma{}}_{22}}{6}-\sqrt{\frac{\left( \tilde{\sigma{}}_{11}-\tilde{\sigma }_{22} \right)^{2}}{2}-\tilde{\sigma{}}_{12}^{2}} \\ \tilde{\sigma{}}_{3}^{\prime}=\frac{\tilde{\sigma }_{11}+\tilde{\sigma{}}_{22}}{3} \end{array} \end{aligned} $$
(10)

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

For multiaxial loading, the yield condition is obtained by replacing the uniaxial stress by an effective stress \(\bar {\sigma {}}\), which lumps in a single quantity all the components of the stress tensor. The yield condition becomes \(\bar {\sigma {}}-\sigma _{R}\left ( \bar {\varepsilon {}} \right )=0\) in which \(\bar {\sigma {}}\) is also called yield function. 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, 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
$$\displaystyle \begin{aligned} d\varepsilon_{pq}=\frac{\partial{\bar{\sigma }}}{\partial{\sigma _{pq}}}d\bar{\varepsilon{}} \end{aligned} $$
(11)
This approach is convenient because it is applicable to either isotropic or anisotropic materials. Therefore, the only question remaining to completely specify the plasticity theory is to define \(\bar {\sigma {}}\), which determines the yield condition and the flow rule.

Isotropic Yield Conditions

Tresca assumed that plastic yielding occurs when the maximum shear stress reaches a critical value, the simple shear flow stress 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
$$\displaystyle \begin{aligned} \bar{\sigma{}}=\sqrt{\frac{\left( \sigma_{1}^{\prime}-\sigma_{2}^{\prime} \right)^{2}+\left( \sigma_{2}^{\prime}-\sigma_{3}^{\prime} \right)^{2}+\left( \sigma _{3}^{\prime}-\sigma_{1}^{\prime} \right)^{2}}{2}} \\ \end{aligned} $$
$$\displaystyle \begin{aligned} &=\sqrt{\frac{\left( \sigma _{1}-\sigma_{2} \right)^{2}+\left( \sigma _{2}-\sigma_{3} \right)^{2}+\left( \sigma _{3}-\sigma_{1} \right)^{2}}{2}}\\ &=\sigma_{R} \end{aligned} $$
(12)
Here, 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

The first anisotropic yield condition was proposed by von Mises (1928) which, when it includes the additional requirement of independence of the mean stress 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
$$\displaystyle \begin{aligned} \bar{\sigma{}}=\sqrt{{F\left( \sigma_{yy}-\sigma _{zz} \right)}^{2}+G\left( \sigma_{zz}-\sigma _{xx} \right)^{2}+H\left( \sigma_{xx}-\sigma _{yy} \right)^{2}+2L\sigma_{yz}^{2}+2M\sigma _{zx}^{2}+{2N\sigma }_{xy}^{2}} =\sigma_{R} \end{aligned} $$
(16)
For sheet forming applications, a plane stress state is very often assumed for solving practical problems. In this case, Hill’s 1948 yield condition reduces to
$$\displaystyle \begin{aligned} \bar{\sigma{}}=\sqrt{\mathrm{F}\sigma_{yy}^{2}+G\sigma_{xx}^{2}+H\left( \sigma_{xx}-\sigma_{yy} \right)^{2}+2N\sigma _{xy}^{2}} =\sigma_{R} \end{aligned} $$
(17)
The resulting associated flow rule is given by applying Eq. (11)
$$\displaystyle \begin{aligned} \begin{aligned} {d\varepsilon }_{xx}=\frac{{\left( G+H \right)\sigma }_{xx}-{H\sigma }_{yy}}{\bar{\sigma }}d\bar{\varepsilon{}} \\ {d\varepsilon }_{yy}=\frac{{\left( H+F \right)\sigma }_{yy}-{H\sigma }_{xx}}{\bar{\sigma }}d\bar{\varepsilon{}} \\ {d\varepsilon }_{zz}=-{d\varepsilon }_{xx}-{d\varepsilon }_{yy}=-\frac{G\sigma_{xx}+{F\sigma }_{yy}}{\bar{\sigma{}}}d\bar{\varepsilon{}} \\ {2d\varepsilon }_{xy}=\frac{{2N\sigma }_{xy}}{\bar{\sigma{}}}d\bar{\varepsilon{}} \end{aligned} {} \end{aligned} $$
(18)
Hill’s 1948 yield condition is the most employed for anisotropic materials. However, it has limitations, in particular, the number of coefficients for plane stress states is only four, which does not always allow a correct description of plastic anisotropy, even in uniaxial tension.

Linear Tensor Transformations

A systematic way to transform convex isotropic yield conditions described by principal (deviatoric) stresses, such as those given in Eqs. (12), (13), (14), and (15), to convex anisotropic yield conditions is to employ the concept of linear transformation initiated in Barlat et al. (1991) and Karafillis and Boyce (1993). In this case, the equations of section “Tensors and Invariants” and the isotropic yield conditions of section “Flow Theory of Plasticity” are fully part of the anisotropic theory. For a pressure-independent material, a linear transformation of the stress deviator is sufficient to describe the anisotropic plastic behavior
$$\displaystyle \begin{aligned} \tilde{\boldsymbol{\upsigma{}}}^{\prime}=\mathbf{C}\boldsymbol{:}\boldsymbol{\upsigma}^{\prime}=\mathbf{C}\boldsymbol{:}\mathbf{T}\boldsymbol{:}\boldsymbol{\upsigma} =\mathbf{L}\boldsymbol{:}\boldsymbol{\upsigma{}} \end{aligned} $$
(19)
where “:” 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 : σ. Each of the fourth-order tensors 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:
$$\displaystyle \begin{aligned} \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \tilde{\sigma{}}_{xx}^{\prime}\\ \tilde{\sigma{}}_{yy}^{\prime}\\ \tilde{\sigma{}}_{zz}^{\prime}\\ \end{array} }\\ {\begin{array}{*{20}c} \tilde{\sigma{}}_{yz}^{\prime}\\ \tilde{\sigma{}}_{zx}^{\prime}\\ \tilde{\sigma{}}_{xy}^{\prime}\\ \end{array} }\\ \end{array} } \right]=\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33}\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{array}}& {\begin{array}{*{20}c} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ C_{44} & 0 & 0\\ 0 & C_{55} & 0\\ 0 & 0 & C_{66}\\ \end{array} }\\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \sigma_{xx}^{\prime}\\ \sigma_{yy}^{\prime}\\ \sigma_{zz}^{\prime}\\ \end{array} }\\ {\begin{array}{*{20}c} \sigma_{yz}^{\prime}\\ \sigma_{zx}^{\prime}\\ \sigma_{xy}^{\prime}\\ \end{array} }\\ \end{array} } \right] \end{aligned} $$
(20)
It is worth noting, though, that all the coefficients C pq , which are different from but related to C pqrs , are not independent.

Non-quadratic Approaches

In order to develop an anisotropic formulation, the principal values of \(\tilde {\boldsymbol {\upsigma {}}}^{\prime }\) substitute the principal deviatoric stresses in an isotropic yield condition as Eq. (13), for instance, leading to It is important to note that the principal values \(\tilde {\sigma {}}_{k}^{\prime }\) must be calculated using Eq. (5) because the invariant \(\tilde {I}_{1}\) is not necessarily 0. Equation (21) combined with a proper orthotropic linear transformation tensor 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).
In general, any isotropic yield function expressed with the principal stresses can be applied to anisotropic behavior by defining a number t of linear transformations similar to Eq. (19)
$$\displaystyle \begin{aligned} \boldsymbol{\upsigma}^{\prime\left( t \right)}={\mathbf{C}}^{\left(t\right)}{\boldsymbol{:\upsigma^{\prime}=}}{\mathbf{C}}^{\left( t \right)}{\boldsymbol{:}\mathbf{T}\boldsymbol{:}\,\boldsymbol{\upsigma{=}}}{\mathbf{L}}^{\left( t \right)}{\boldsymbol{:}\boldsymbol{\upsigma{}}} \end{aligned} $$
(22)
with suitable anisotropy coefficient tensors. With two linear transformations, i.e., 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

Alternatively, specific formulations were also developed for plane stress cases, which are useful in simple analysis or in FE simulations of sheet forming processes using shell elements. For instance, the so-called Yld2000-2d yield function (Barlat et al., 2003a) is also a generalization of Hershey’s isotropic yield function. Note that this yield condition was proposed under different forms in the literature (Aretz, 2004; Banabic et al., 2005) as pointed out by Barlat et al. (2007). The first step is to recognize that for plane stress, \(\sigma _{1}=2{\sigma ^{\prime }_1}+{\sigma ^{\prime }_2}\) and \(\sigma _{2}=2{\sigma ^{\prime }_2}+{\sigma ^{\prime }_1}\). Therefore, Eq. (13) reduces to with
$$\displaystyle \begin{aligned} &\phi =\frac{1}{2}\left| {\sigma^{\prime}_1}-{\sigma^{\prime}_2} \right|{}^{a} \end{aligned} $$
(25)
$$\displaystyle \begin{aligned} &\psi =\frac{1}{2}\left\{ \left| {2\sigma^{\prime}_2}+{\sigma^{\prime}_1} \right|{}^{a}+\left| {2\sigma^{\prime}_1}+{\sigma^{\prime}_2} \right|{}^{a} \right\} \end{aligned} $$
(26)
Because a plane stress state can be described by two principal values only, 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.,
$$\displaystyle \begin{aligned} {2\bar{\sigma{}}}^{a}&=\left| \sigma_{1}^{\prime\left( 1 \right)}-\sigma _{2}^{\prime\left( 1 \right)} \right|{}^{a}+\left| 2\sigma_{2}^{\prime\left( 2 \right)}+\sigma_{1}^{\prime\left( 2 \right)} \right|{}^{a}\\&+ \left| 2\sigma _{1}^{\prime\left( 2 \right)}+\sigma_{2}^{\prime\left( 2 \right)} \right|{}^{a}=2\sigma _{R}^{a} \end{aligned} $$
(27)
For plane stress, the matrix representation of the tensor to deviator transformation is and, because it is implicitly assumed that \(\sigma _{zz}^{\prime \left ( t \right )}={-\sigma }_{xx}^{\prime \left ( t \right )}-\sigma _{yy}^{\prime \left ( t \right )}\), the matrix representation of the linear transformations in Eq. (22) reduces to
$$\displaystyle \begin{aligned} \left[ {\begin{array}{*{20}c} \sigma_{xx}^{\prime\left( t \right)}\\ \sigma_{yy}^{\prime\left( t \right)}\\ \sigma_{xy}^{\prime\left( t \right)}\\ \end{array} } \right]=\left[ {\begin{array}{*{20}c} C_{11}^{\left( t \right)} & C_{12}^{\left( t \right)} & 0\\ C_{21}^{\left( t \right)} & C_{22}^{\left( t \right)} & 0\\ 0 & 0 & C_{66}^{\left( t \right)}\\ \end{array} } \right]\left[ {\begin{array}{*{20}c} \sigma_{xx}^{\prime}\\ \sigma_{yy}^{\prime}\\ \sigma_{xy}^{\prime}\\ \end{array} } \right] \end{aligned} $$
(29)

Anisotropy Coefficients

The principal values of \({{\boldsymbol {\upsigma {}}}}^{\prime {\left ( t \right )}}\) can be calculated with Eqs. (9) or (10). The yield condition of Eq. (27) reduces to the isotropic expression when the matrices \({\mathbf {C}}^{\left ( t \right )}\) are both taken as the identity matrix so that \({{\boldsymbol {\upsigma {}}}}^{\prime {\left ( 1 \right )}}={{\boldsymbol {\upsigma {}}}}^{\prime \left ( 2 \right )}={\boldsymbol {\upsigma {}}}^{\prime }\). For all stress states where the shear stress is zero, the principal values of \({{\boldsymbol {\upsigma {}}}}^{\prime \left ( k \right )}\) are
$$\displaystyle \begin{aligned} \sigma_{xx}^{\prime\left( t \right)}=C_{11}^{\left( t \right)}\sigma _{xx}^{\prime}+C_{12}^{\left( t \right)}\sigma_{yy}^{\prime} \\ {} \sigma_{yy}^{\prime\left( t \right)}=C_{21}^{\left( t \right)}\sigma _{xx}^{\prime}+C_{22}^{\left( t \right)}\sigma_{yy}^{\prime} \end{aligned} $$
(30)
Then, the yield function in Eq. (27) becomes
$$\displaystyle \begin{aligned} \begin{array}{l} {2\bar{\sigma{}}}^{a}{=\left| {\left( C_{11}^{\left( 1 \right)}-C_{21}^{\left( 1 \right)} \right)}\sigma_{xx}^{\prime}-{\left( C_{12}^{\left( 1 \right)}-C_{22}^{\left( 1 \right)} \right)}\sigma _{yy}^{\prime} \right|}^{a} +\left| {\left( {2C}_{21}^{\left( 2 \right)}+C_{11}^{\left( 2 \right)} \right)}\sigma _{xx}^{\prime}+ \right.\\ \left.{\left( {2C}_{22}^{\left( 2 \right)}+C_{12}^{\left( 2 \right)} \right)}\sigma _{yy}^{\prime} \right|{}^{a}+\left| {\left( {2C}_{11}^{\left( 2 \right)}+C_{21}^{\left( 2 \right)} \right)}\sigma _{xx}^{\prime}+{\left( {2C}_{12}^{\left( 2 \right)}+C_{22}^{\left( 2 \right)} \right)}\sigma _{yy}^{\prime} \right|{}^{a} \end{array} \end{aligned} $$
(31)
Because the yield function depends on \(C_{11}^{\left ( 1 \right )}-C_{21}^{\left ( 1 \right )}\) and \(C_{12}^{\left ( 1 \right )}-C_{22}^{\left ( 1 \right )}\) for the first transformation, only three coefficients are independent in \({\mathbf {C}}^{\left ( 1 \right )}\). Therefore, in this work, the condition \(C_{12}^{\left ( 1 \right )}=C_{21}^{\left ( 1 \right )}=0\) is imposed for convenience, but it is worth noting that \(C_{12}^{\left ( 1 \right )}=C_{21}^{\left ( 1 \right )}=1\) would be an acceptable condition. Equation (31) can also be expressed as
$$\displaystyle \begin{aligned} {2\bar{\sigma{}}}^{a}&=\left| {\alpha_{1}\sigma }_{xx}^{\prime}-{\alpha_{2}\sigma }_{yy}^{\prime} \right|{}^{a} +\left| {\alpha_{3}\sigma }_{xx}^{\prime}+{2\alpha_{4}\sigma }_{yy}^{\prime} \right|{}^{a}\\&+\left| {{2\alpha }_{5}\sigma }_{xx}^{\prime}+{\alpha_{6}\sigma }_{yy}^{\prime} \right|{}^{a} \end{aligned} $$
(32)
with the following notations:
$$\displaystyle \begin{aligned} \begin{aligned} \alpha_{1}&=C_{11}^{\left( 1 \right)}-C_{21}^{\left( 1 \right)}=C_{11}^{\left( 1 \right)}\\ \alpha_{2}&=C_{22}^{\left( 1 \right)}-C_{12}^{\left( 1 \right)}=C_{22}^{\left( 1 \right)}\\ \alpha_{3}&={2C}_{21}^{\left( 2 \right)}+C_{11}^{\left( 2 \right)}\\ \alpha_{4}&={2C}_{22}^{\left( 2 \right)}+C_{12}^{\left( 2 \right)}\\ \alpha_{5}&={2C}_{11}^{\left( 2 \right)}+C_{21}^{\left( 2 \right)}\\ \alpha_{6}&={2C}_{12}^{\left( 2 \right)}+C_{22}^{\left( 2 \right)} \end{aligned} {}\end{aligned} $$
(33)
When only the shear stress does not vanish, the principal values of \({{\boldsymbol {\upsigma {}}}}^{\prime \left ( 1 \right )}\) and \({{\boldsymbol {\upsigma {}}}}^{\prime {\left ( 2 \right )}}\) are
$$\displaystyle \begin{aligned} \sigma_{k}^{\prime\left( t \right)}=\pm C_{66}^{\left( t \right)}\sigma _{xy} \end{aligned} $$
(34)
with positive coefficients \(C_{66}^{\left ( t \right )}\). The yield function reduces to
$$\displaystyle \begin{aligned} {2\bar{\sigma{}}}^{a}&=\left\{ {2^{a}\left( C_{66}^{\left( 1 \right)} \right)}^{a}+2\left( C_{66}^{\left( 2 \right)} \right)^{a} \right\}\left| \sigma_{xy} \right|{}^{a}\\&=\left\{ 2^{a}\alpha_{7}^{a}+2\alpha _{8}^{a} \right\}\left| \sigma_{xy} \right|{}^{a} \end{aligned} $$
(35)
where
$$\displaystyle \begin{aligned} \begin{aligned} \alpha_{7}&=C_{66}^{\left( 1 \right)} \\ \alpha_{8}&=C_{66}^{\left( 2 \right)} \end{aligned}{} \end{aligned} $$
(36)

Anisotropy Coefficient Tensors

Inverting the relationship between the coefficients α j and \(C_{mn}^{\left ( t \right )}\) in Eqs. (33) and (36), the matrix representations of tensors \({\mathbf {C}}^{\left ( t \right )}\) become
$$\displaystyle \begin{aligned} \begin{aligned} \left[ {\mathbf{C}}^{\left( 1 \right)} \right]=\left[ {\begin{array}{*{20}c} \alpha_{1} & 0 & 0\\ 0 & \alpha_{2} & 0\\ 0 & 0 & \alpha_{7}\\ \end{array} } \right] \\ \left[ {\mathbf{C}}^{\left( 2 \right)} \right]=\frac{1}{3}\left[ {\begin{array}{*{20}c} {4\alpha }_{5}-\alpha_{3} & 2\left( \alpha_{6}-\alpha_{4} \right) & 0\\ 2\left( \alpha_{3}-\alpha_{5} \right) & {4\alpha }_{4}-\alpha_{6} & 0\\ 0 & 0 & {3\alpha }_{8}\\ \end{array} } \right] \end{aligned}{} \end{aligned} $$
(37)
where all the independent coefficients 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:
$$\displaystyle \begin{aligned} \begin{aligned} \left[{\mathbf{L}}^{\left( 1 \right)} \right]&=\frac{1}{3}\left[ {\begin{array}{*{20}c} {2\alpha }_{1} & -\alpha_{1} & 0\\ -\alpha_{2} & {2\alpha }_{2} & 0\\ 0 & 0 & {3\alpha }_{7}\\ \end{array} } \right] \\ \left[ {\mathbf{L}}^{\left( 2 \right)} \right]&=\frac{1}{9}\left[ {\begin{array}{*{20}c} {8\alpha }_{5}-{2\alpha }_{3}-2\alpha_{6}+2\alpha_{4} & {4\alpha }_{6}-{4\alpha }_{4}-4\alpha_{5}+\alpha_{3} & 0\\ {4\alpha }_{3}-{4\alpha }_{5}-4\alpha_{4}+\alpha_{6} & {8\alpha }_{4}-{2\alpha }_{6}-2\alpha_{3}+2\alpha_{5} & 0\\ 0 & 0 & {9\alpha }_{8}\\ \end{array} } \right] \end{aligned}{} \end{aligned} $$
(38)
In addition to the exponent 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

The associated flow rule, Eq. (11) can be obtained from chain rule derivation
$$\displaystyle \begin{aligned} d\varepsilon_{pq}=\frac{\partial{\bar{\sigma }}}{\partial{\sigma _{pq}}}d\bar{\varepsilon{}}=\left\{ \frac{\partial{\bar{\sigma }}}{\partial \sigma_{k}^{\prime\left( 1 \right)}}\frac{{\partial{\sigma }}_{k}^{\prime\left( 1 \right)}}{\partial{\sigma_{rs}^{\prime\left( 1 \right)}}}L_{rspq}^{\left( 1 \right)}+\frac{\partial{\bar{\sigma }}}{\partial{\sigma_{k}^{\prime\left( 2 \right)}}}\frac{{\partial{\sigma }}_{k}^{\prime\left( 2 \right)}}{\partial{\sigma _{rs}^{\prime\left( 2 \right)}}}L_{rspq}^{\left( 2 \right)} \right\}d\bar{\varepsilon{}} \end{aligned} $$
(39)
since
$$\displaystyle \begin{aligned} \frac{{\partial{\sigma }}_{rs}^{\prime\left( t \right)}}{\partial{\sigma _{pq}}}=L_{rspq}^{\left( t \right)} \end{aligned} $$
(40)
Note that there are two singular cases in the expression of 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

Plane strain tension, for which the strain in one direction of the sheet is equal to zero, is one of the most typical deformation modes occurring in press formed sheet metal parts, and is also the dominant deformation mode in a localized neck of sheet metals. Figure 1 shows a typical specimen geometry used for a plane strain tensile test. When a tensile force is applied to the specimen, the deformation is approximately plane strain in the central part of the gauge section but, closer to the free surface, it gradually changes to uniaxial tension. In order to exclude the effect of the nonuniform deformation on the accuracy of stress measurement for the plane strain tension test, the same tensile plastic strain \(\varepsilon _{ps}^{p}\) is applied to two specimens with different gauge widths W, and the difference in the tensile force ΔF is measured. Thus, the flow stress under plane strain tension, σ ps , is calculated as
$$\displaystyle \begin{aligned} \sigma_{ps}=\frac{\varDelta{F}}{\varDelta{W}}\times \frac{\exp{{(\varepsilon _{ps}^{p})}}}{t_{0}} \end{aligned} $$
(41)
where 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/(2R) ≥ 12 so that plane strain tension is maintained in the central part of the gauge section (An et al., 2004).
Fig. 1

Specimen for plane strain tension test (An et al., 2004). The shaded areas are clamping areas. W: width of the gauge section; R: radius of notch tip

An alternative approach to post-processing the plane strain tension test is to use full-field strain measurement methods (such as Digital Image Correlation, or DIC) and finite element analysis. The DIC results corresponding to the specimen geometry in Fig. 2a are shown in Fig. 2b. These figures confirm that plane strain conditions prevail at the central region of the specimen, throughout the deformation. In designing the specimen, a compromise must be reached between making the width of the gauge section wide enough so that plane strain conditions develop over a sufficient area at the center but also long enough in the loading direction so that DIC can be used with confidence. Subsequently, a finite element model of the specimen is created, adopting at first the von Mises yield criterion. The model is used to probe the stress field at the center of the specimen. The stress along the loading direction, i.e., 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.
Fig. 2

(a) Geometry of a plane strain tension specimen (Tian et al., 2017, adapted from Tardif and Kyriakides, 2012). (b) Measurement of strain field as the plastic work W P accumulates during the test, revealing that plane strain conditions prevail in the central 50% of the gauge section width (With permission of Springer)

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

The hydraulic bulge test is widely used in determining the work hardening characteristics of sheet materials at plastic strains greater than those achieved in simple tension. Figure 3 shows a schematic of the hydraulic bulge test. The hydraulic pressure is applied to one side of a circular sheet specimen and the development of the radius of curvature 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  = /(2t) 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.
Fig. 3

Hydraulic bulge test (a) Schematic of the testing machine, the bulge specimen and the sensors to measure curvature and apex thickness. (b) Schematic of the bulging specimen at the apex, showing the equibiaxial tension condition

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

In this testing method (Fig. 4), two independent normal stress components are applied to two of the three principal directions of a cubic specimen made of adhesively bonded sheet laminate, to measure the yield locus of a sheet sample in the 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).
Fig. 4

Biaxial compressive test using an adhesively bonded sheet laminate specimen. (a) Schematic representation of the experimental method to determine a yield locus in the normal stress plane. (b) Yield loci (contours of the von Mises equivalent plastic strain, at ε =0.002) for steel (S10C) prestretched by various strains. (Tozawa, 1978, with permission of Springer)

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).

Figure 5 shows the shape and dimensions of the cruciform specimen specified in ISO (2014b). The minimum requirements for the specimen geometry are as follows:
Fig. 5

Cruciform specimen (ISO (2014b)). 1: gauge area; 2: arm; 3: grip; 4: slit; B Sx and B Sy : distance between opposing slit ends; B: arm width; C: grip length; L: slit length; w S: slit width. : Optimum positions for strain measurement (when the maximum principal stress direction is parallel to the x −axis)

  1. (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.

     
  2. (b)

    Seven slits per one arm shall be made. The slit length L should be in the range B ≤ L ≤ 2B. The slit width w S should be made as small as possible, and should be less than 0.01B.

     
  3. (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 ≤ 0.08B 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

Figure 6 shows a schematic diagram of the combined tension-compression test and specimen geometry (Kuwabara et al., 2007). The lower dies 1 and 2 are fixed to the chucks of a biaxial tensile testing machine, so that the teeth of each of the lower dies are meshing with each other. A specimen is set on the lower dies as shown in the figure. Compression platens fixed to the lower dies apply compressive force to the sides of the gauge area of the specimen. The upper dies 1 and 2 are placed over the specimen and fixed to the lower dies using bolts with a gap of 101% of the sheet thickness between the upper and lower dies. Teflon sheets are inserted between the specimen and the dies to reduce the friction between the dies and specimen. The strain components of the specimen are measured using a biaxial strain gauge mounted on the center of the gauge area.
Fig. 6

Testing jigs and specimen for combined tension-compression test (Kuwabara et al., 2007). (a) Specimen sandwiched between the lower and upper dies with both ends clamped by the chucks of a biaxial tensile testing machine. (b) Geometry of the specimen used for the combined tension-compression test (dimensions in mm)

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.

Figure 7a shows the comb-type dies developed for application of continuous in-plane stress reversal to a sheet specimen (Kuwabara et al., 2009). A sheet specimen is put between the upper and lower die pairs and subjected to continuous in-plane stress reversals under a constant blank-holding force to prevent buckling. The appropriate magnitude of the blank-holding pressure is approximately 1% of the tensile strength of the material. It is recommended to apply lubrication on both sides of the specimen with Vaseline and Teflon sheets to prevent the specimen from galling the dies. A strain gauge is used to measure the strain. According to the FE analysis of the specimen deformation and the strain measurement position shown in Fig. 7b, the stress measurement error is estimated to be less than 1% (Noma and Kuwabara, 2012). Other types of stress reversal testing methods are described in Kuwabara (2007).
Fig. 7

Experimental apparatus for application of in-plane stress reversals to a sheet specimen; (a) configuration of the dies (Kuwabara et al., 2009) and (b) geometry of the specimen (Noma and Kuwabara, 2012)

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.

Figure 8 shows a schematic diagram of the multiaxial tube expansion testing method. An axial load and internal pressure are applied to a tubular specimen, and both are servo-controlled using an electrical, closed-loop control system, so that control of the true stress path is possible. During the test, it is possible for the load and pressure to drop, e.g., after the specimen has experienced a limit load instability. The deformation behavior of a specimen can be measured using strain gauges and a spherometer (Kuwabara et al., 2005), strain gauges and extensometers (Korkolis and Kyriakides, 2008), displacement meters (Kuwabara and Sugawara, 2013), or a digital image correlation system (Hakoyama and Kuwabara, 2015; Ripley and Korkolis, 2016).
Fig. 8

Schematic diagram of multiaxial tube expansion testing method

The axial and circumferential membrane strains, ε ϕ and ε θ , are evaluated at midthickness using the following equations:
$$\displaystyle \begin{aligned} \begin{array}{rcl} {}&\varepsilon_{\phi{}}=\varepsilon_{\phi \mathrm{S}}\mathrm{-ln}\frac{R_{\phi{}}}{R_{\phi }\mathrm{-(}t\mathrm{/2)}}\mathrm{,}&\\ &\varepsilon_{\theta }=\mathrm{ln}\frac{D\mathrm{-}t}{D_{\mathrm{0}}\mathrm{-}t_{\mathrm{0}}}=\mathrm{ln}\frac{D_{\mathrm{0}}\mathrm{exp(}\varepsilon _{\theta{\mathrm{S}}}\mathrm{)-}t}{D_{\mathrm{0}}\mathrm{-}t_{\mathrm{0}}}&\end{array} \end{aligned} $$
(42)
where 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
$$\displaystyle \begin{aligned} t=\frac{D}{\mathrm{2}}\mathrm{-}\sqrt{\left( \frac{D}{\mathrm{2}} \right)^{\mathrm{2}}\mathrm{-}\left( \frac{\mathrm{(}D_{\mathrm{0}}\mathrm{-}t_{\mathrm{0}}\mathrm{)}t_{\mathrm{0}}}{\mathrm{exp(}\varepsilon _{\phi{}}\mathrm{)}} \right)} \end{aligned} $$
(43)
The axial and circumferential true stress components, 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:
$$\displaystyle \begin{aligned} \sigma_{\phi{}}=\frac{P\pi (D/2-t)^{2}+T}{\pi (D-t)t} \end{aligned} $$
(44)
$$\displaystyle \begin{aligned} \sigma_{\theta{}}=\frac{(R_{\phi}-t)(D-2t)}{(2R_{\phi}-t)t}P-\frac{D-t}{2R_{\phi} -t}\sigma_{\phi} \end{aligned} $$
(45)
where 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

In this section, an example of material modeling using the Yld2000-2d yield function is presented for a low-carbon steel sheet sample. For this material, uniaxial tension (section “Uniaxial Tension Test”), in-plane biaxial tension (section “Biaxial Tensile Test Using a Cruciform Specimen”), and multiaxial tube expansion (section “Multiaxial Tube Expansion Test”) test data are available. All experiments were performed on the same sheet lot. In what follows, different calibration techniques are performed utilizing the uniaxial data mainly for identification, and the resulting yield loci are compared with those experimentally determined. The anisotropic parameters α i (i = 1 − 8) and exponent a of the Yld2000-2d yield function are determined to minimize the cost function given by
$$\displaystyle \begin{aligned} F=\sum\limits_{j=1}^N {w_{j,l}\left(l^{\prime} -l \right)^{2}} +\sum\limits_{j=1}^N w_{j,\beta}\left(\beta^{\prime}-\beta \right)^{2} \end{aligned} $$
(46)
where 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.
Two methods, A and B, are adopted to evaluate the cost function 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.
Table 1

Values of w j,l and w j,β used for the methods A and B

 

Method A

Method B

N

7

8

Tensile direction from RD θ/

w j,l

w j,β

w j,l

w j,β

 

1000

0.2

1000

0.2

15, 30, 45, 60, 75, 90

1

0.2

1

0.2

Equibiaxial

NA

NA

1

0.2

The calculated values of 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 \(\varepsilon _{0}^{p}\).
Table 2

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

 

Initial a

a

α 1

α 2

α 3

α 4

α 5

α 6

α 7

α 8

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

Figure 9a compares the Yld2000-2d yield loci determined from the methods A-1, A-2, A-3, and B with the experimental work contours associated with reference plastic strains \(\varepsilon _{0}^{p}\). The shapes of the work contours significantly change with increasing \(\varepsilon _{0}^{p}\) for a strain range of \(\varepsilon _{0}^{p}\le 0.08\), while it converges into a similar shape for \(\varepsilon _{0}^{p}> 0.08\) to be consistent with the Yld2000-2d yield locus determined from the method B. On the other hand, the shapes of the yield loci based on the method A significantly depend on the initial value of 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.
Fig. 9

Experimental results of the biaxial tensile tests on a cold rolled ultralow-carbon steel sheet. (a) Measured stress points forming contours of plastic work, compared with the yield loci calculated using the Yld2000-2d yield function. Each symbol corresponds to a work contour associated with a particular value of the reference plastic strain \(\varepsilon _{\mathrm {0}}^{\mathrm {p}}\). The stress values forming a work contour were normalized by σ 0, the uniaxial tensile true stress in the RD associated with \(\varepsilon _{\mathrm {0}}^{\mathrm {p}}\), belonging to the work contour. (b) The directions of plastic strain rates measured at different levels of plastic work per unit volume, represented by \(\varepsilon _{\mathrm {0}}^{\mathrm {p}}\), compared with those calculated using the Yld2000-2d yield function

Figure 10 compares (a) the uniaxial tensile flow stresses and (b) the r-values with those predicted by the Yld2000-2d yield function. The four methods all give a fair agreement with the measurement. Consequently, it is concluded that the Yld2000-2d yield function determined by method B gives the closest agreement with the experimental data for both the work contours and the directions of the plastic strain rates, as well as for the anisotropy in tensile flow stresses and r-values.
Fig. 10

Experimental (a) normalized flow stress and (b) r-value directionalities compared with those calculated from the Yld2000-2d yield function

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.

The flow stresses from the 7 UT tests and their evolution with plastic deformation are shown in Fig. 11a. During the UT experiments, the strain field was captured using DIC. This data was then used to calculate the plastic strain ratios (r-value), which are shown in Fig. 11b. The first two curves in that figure are shown with dashed lines, due to noise in the DIC measurements in the immediate post-yield regime. The results of these two figures indicate that the sheet is anisotropic and that the anisotropy evolves in the small strain, post-yield regime, almost stabilizing past 4.2 MJ/m3 (which corresponds to a strain of 2.5% in uniaxial tension).
Fig. 11

Evolution of (a) the flow stress and (b) the plastic strain ratio at every 15o to the RD, with increasing plastic work per unit volume (Tian et al., 2017 With permission of Springer)

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.

The results from the UT, PST, and DC tests are used to calibrate the Yld2000-2d anisotropic yield function. There are numerous ways to perform the fit, e.g., as described in section “Application to Steel”. Two readily implemented approaches involve minimization of a cost function using nonlinear least squares fit or Newton-Raphson. The former tends to provide easier convergence and a better fit in an average sense, while the latter tends to be more powerful and provides a better agreement with the individual data points, though it can be harder to converge. For the present material, the exponent 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.
Fig. 12

Agreement between anisotropic yield criterion Yld2000-2d and experiments. (a) Yield locus, including isoshear contours, at Wp = 20 MJ/m3 and; (b) Plastic strain ratios (Tian et al., 2017 With permission of Springer)

Table 3

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

a

α 1

α 2

α 3

α 4

α 5

α 6

α 7

α 8

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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Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  • Frédéric Barlat
    • 1
    Email author
  • Toshihiko Kuwabara
    • 2
  • Yannis P. Korkolis
    • 3
  1. 1.Graduate Institute of Ferrous TechnologyPohang University of Science and TechnologyPohangRepublic of Korea
  2. 2.Division of Advanced Mechanical Systems Engineering, Institute of Engineering (double as Institute of Global Innovation Research)Tokyo University of Agriculture and TechnologyTokyoJapan
  3. 3.Department of Mechanical EngineeringUniversity of New HampshireDurhamUSA

Section editors and affiliations

  • Sergey Alexandrov
    • 1
  1. 1.Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia