Yield Criteria

Univ.-Prof. Dr.-Ing. habil. Josef BETTEN RWTH Aachen University Mathematical Models in Materials Science and Continuum Mechanics Augustinerbach 4-20 D-52056 A a c h e n , Germany

<betten@mmw.rwth-aachen.de>

Abstract

This worksheet is concerned with the formulation of yield criteria for isotropic and anisotropic materials. Yielding of anisotropic materials can be characterized by yield criteria, which are scalar-valued functions of the stress tensor and of material tensors, for instance, of rank two or four, characterizing the anisotropic properties of the material. Because of the requirement of invariance, a yield criterion can be expressed as a single-valued function of the integrity basis. In finding an integrity basis involving the stress tensor and material tensors, the constutive equations are first formulated based on the tensor function theory. Since the plastic work characterizes the yield process, we read from this scalar expression the essential invariants to formulate a yield criterion. Some examples for practical use have been discussed in more detail.

Keywords: Plastic Yielding; Isotropic and Anisotropic Materials; Plastic Work; Integrity Basis; Incompressibility & Compressibility; Strength Differential Effect;

Introduction

For the analysis of strain and stress distributions in a material loaded beyond the elastic limit, a constitutive theory of plasticity must specify the yield condition under multiaxial states of stress, since the uniaxial condition sigma = Y, where Y is the yield stress, is inadequate if there is more than one stress component sigma. Furthermore, the theory of plasticity must specify the postyield behavior. Thus, the following questions arise:

1) Yield Criterion. What stress combinations cause plastic deformations?

2) Postyield Behavior. How are the plastic deformation increments related to the stress components? How does the yield condition change with workhardening?

A yield condition for a material under multiaxial states of stress can be seen as a law defining the limit of elasticity under any possible combination of stresses. A yield condition is a special form

1

of a failure criterion, which indicates the onset of plastic deformation.

Formulation of Yield Criteria

A yield condition for a material under multiaxial states of stress can be geometrically interpreted as a limiting envelope in the stress space, and yielding occurs when the given stress vector (not to be confused with the term traction) penetrates the yield surface in the stress space. The general form of a yield criterion is given by a scalar-valued tensor function > restart:> f(sigma[ij],A[ij],A[ijkl])=1;

= ( )f , ,σij Aij Aijkl 1of the Cauchy stress tensor sigma and several material tensors A of rank two and four, for instance, characterizing the anisotropy of the material. Due to the rquirement of invariance under any orthogonal transformation, the function f in the above criterion can be expressed as a single-valued function of the integrity basis, the elements of which are the irreducible invariants. Together with the invariants of the single argument tensors, the set of simutaneous or joint invariants forms the integrity basis. The theory of invariants for several argument tensors of rank two has been developed, for instance, by RIVLIN (1970) and SPENCER (1971, 1987). Integrity basis of an order higher than two has been dicussed by BETTEN (1982, 1987, 1993, 1998). To formulare a yield criterion f (...) = 1 it is not necessary to consider the complete set of ireducible invariants. In finding the essential invariants, we first formulate the constitutive equation > restart: macro(eps=epsilon,sig=sigma,pl=plastic):> d*eps[ij][pl]=f[ij](sig[pq],A[pq],A[pqrs]);

= d εijplastic

( )fij , ,σpq Apq Apqrs

based upon the representation theory of tensor-valued functions. On the left hand side we have the plastic strain increments. Since the plastic work > d*W=sig[ji]*d*eps[ij][pl];

= d W σji d εijplastic

characterizes the yield process, we read from its scalar expression the relevant invariants to formulate a yield criterion, as has been pointed out by BETTEN (1988) in more detail. Note, on the right hand side and in the following text EINSTEIN 's summation convention has been utilized. Constitutive equations describing plastic deformation can be formulated, for instance, in two different ways as pointed out in the following. In the classical theory of plasticity the flow rule > restart: macro(sig=sigma,eps=epsilon,pl=plastic):> d*eps[ij][pl]=Diff(F,sig[ij])*d*lambda;

= d εijplastic

⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂σij

F d λ

is used, where the factor d*lambda can be interpreted as a LAGRANGE multiplier, as illustrated by BETTEN (2001). The scalar function F represents the plastic potential. Based upon this flow rule and assuming stable or indifferently stable materials (DRUCKER's postulate), the furface

2

F = const. should be convex (BETTEN, 1979, 2001).

Isotropic Materials

From the theory of isotropic tensor functions discussed by SPENCER (1971), it is evident that for isotropic medium the scalar function F might be expressed as a single-valued function of the irreducible basic invariants > [S[1],S[2],S[3]]=

[sig[kk],sig[ik]*sig[ki],sig[ij]*sig[jk]*sig[ki]];

= [ ], ,S1 S2 S3 [ ], ,σkk σik σki σij σjk σki

of the stress tensor. Thus, considering the plastic potential > F(sig[ij]):=F(S[nu]); # nu=1,2,3

:= ( )F σij ( )F Sν

and using the classical flow rule, one immediately obtains the following constitutive equation: > restart: macro(eps=epsilon,sig=sigma,pl=plastic,la=lambda):> d*eps[ij][pl]=(Diff(F,S[1])*delta[ij]+

2*Diff(F,S[2])*sig[ij]+3*Diff(F,S[3])*sig[ik]*sig[kj])*d*la;

= d εijplastic

⎛

⎝⎜⎜

⎞

⎠⎟⎟ + +

⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂S1

F δij 2⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂S2

F σij 3⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂S3

F σik σkj d λ

Instead of the plastic potential theory, one can represent the strain incremental tensor as a tensor-valued function of one second rank argument tensor: > d*eps[ij][pl]=f[0]*delta[ij]+f[1]*sig[ij]+f[2]*sig[ik]*sig[kj];

= d εijplastic

+ + f0 δij f1 σij f2 σik σkj

The plastic work is given by > d*W =

f[0]*sig[kk]+f[1]*sig[ik]*sig[ki]+f[2]*sig[ij]*sig[jk]*sig[ki];

= d W + + f0 σkk f1 σik σki f2 σij σjk σki

from which we read the three basic invariants defined above. The three scalar-valued functions f [0] ,..., f [2] depend on the integrity basis. Comparing the two constitutive equations, we arrive at the following identities: > restart: macro(la=lambda):> [f[0],f[1],f[2]]=

[Diff(F,S[1]),2*Diff(F,S[2]),3*Diff(F,S[3])]*d*la;

= [ ], ,f0 f1 f2⎡

⎣⎢⎢

⎤

⎦⎥⎥, ,

∂∂S1

F 2⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂S2

F 3⎛

⎝⎜⎜

⎞

⎠⎟⎟∂

∂S3

F d λ

By eliminating the plastic potential F , one can impose additional restrictions on the above scalar-functions, if the existence of a plastic potential is assumed. This problem was solved by BETTEN (1985). Thus, in the isotropic special case, the plastic potential theory is compatible with the tensor functtion theory if some conditions of integrability have been fulfilled. However, for anisotropic materials the plastic potential theory provides only restricted forms of constitutive equations, even if a general plastic potential has been assumed. Consequently, the classical flow rule must be modified for anisotropic solids. Appropriate modifications have been discussed and

3

the resulting conditions of compatibility were derived by BETTEN (1985).

Anisotropic Materials

First let us consider oriented solids. To describe yielding of such materials, one can utilized the tensor generator A = v * v , where the symbol (*) indicates the dyadic product, which can be expressed in index notation as: > restart: > A[ij]:=v[i]*v[j];

:= Aij vi vj

The vector v specifies a privileged direction (transverse isotropy). The restrictions imposed by the symmetries and orientation are then automatically satisfied, and the yield function is a scalar-valued function of two symmetric second-order tensors: > restart: macro(sig=sigma,Om=Omega):> f = f(sig[ij],A[ij]);

= f ( )f ,σij Aij

the representation of which is given in the usual manner, for instance, by SPENCER (1971): > f=f(S[nu],T[nu],Om[1]...Om[4]);

= f ( )f , ,Sν Tν .. Ω1 Ω4

where the ten irreducible invariants > [S[nu]]=[sig[kk],sig[ik]*sig[ki],sig[ij]*sig[jk]*sig[ki]];

= [ ]Sν [ ], ,σkk σik σki σij σjk σki

> [T[nu]]=[A[kk],A[ik]*A[ki],A[ij]*A[jk]*A[ki]];

= [ ]Tν [ ], ,Akk Aik Aki Aij Ajk Aki

> [Om[1]...Om[4]]=[sig[ik]*A[ki],sig[ij]*A[jk]*A[ki], A[ij]*sig[jk]*sig[ki],sig[ij]*sig[jk]*A[kl]*A[li]];

= [ ] .. Ω1 Ω4 [ ], , ,σik Aki σij Ajk Aki Aij σjk σki σij σjk Akl Ali

form the integrity basis. In order to find the relevant invariants we first formulate the constitutive eqution involving two symmetric argument tensors as a minimal polynomial function: > restart: macro(sig=sigma,eps=epsilon,la=lambda,pl=plastic):> d*eps[ij][pl]=(1/2)*Sum(Sum(Phi[la,nu]*(sig[ik]^la*A[kj]^nu+

A[ik]^nu*sig[kj]^la),la=0..2),nu=0..2);

= d εijplastic

12

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟∑

= ν 0

2 ⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟∑

= λ 0

2

Φ ,λ ν ( ) + σikλ

Akjν

Aikν

σkjλ

Since the plastic work characterizes the yield process, we read from its expression the relevant invariants to formulate a yield criterion: > d*W=sig[ji]*rhs(%);

= d W12

σji

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟∑

= ν 0

2 ⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟∑

= λ 0

2

Φ ,λ ν ( ) + σikλ

Akjν

Aikν

σkjλ

> d*W = Sum(Sum(Phi[la,nu]*A[ik]^nu*sig[ki]^(la+1),

4

la=0..2),nu=0..2);

= d W ∑ = ν 0

2 ⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟∑

= λ 0

2

Φ ,λ ν Aikν

σki( ) + λ 1

> relavant_invariants:=A[ik]^nu*sig[ki]^(la+1);

:= relavant_invariants Aikν

σki( ) + λ 1

Note that lambda should be less than two, if nu is not equal to zero. Thus, we find instead of ten only seven relevant invariants: > with(linalg):> A[jk]^nu*sig[kj]^(la+1)=matrix(3,3,[S[1],S[2],S[3],

Omega[1],Omega[3],_,Omega[2],Omega[4],_]);>

= Ajkν

σkj( ) + λ 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

S1 S2 S3

Ω1 Ω3 _Ω2 Ω4 _

Furthermore, without loss of generality in the case of incipient motion (onset of plastic deformation) the vector v can be regarded as a unit vector in the reference configuration. Then, we find a further reduction from seven to five relevant invariants: > A[ik]^nu*sig[ki]^(la+1)=

matrix(2,3,[S[1],S[2],S[3],Omega[1],Omega[3],_]);

= Aikν

σki( ) + λ 1 ⎡

⎣⎢⎢

⎤

⎦⎥⎥

S1 S2 S3

Ω1 Ω3 _Thus, the yield condition is given by: > F(S[1]..S[3],Omega[1],Omega[3])=1;

= ( )F , , .. S1 S3 Ω1 Ω3 1A more general case of anisotropy is described by a material tensor of rank four discussed by BETTEN (1988, 1998, 2001), for instance.

Examples

In the following some simple examples for practical use should be discussed. The first example is based upon a plastic potential consisting of a linear combination of the two deviatoric invariants. Thus, we assume isotropy, incompressibility, and different behavior in tension and compression (strength differential effect): > restart: > F=`J'`[2]+alpha*`J'`[3]/Y;

= F + J'2α J'3

Ywhere Y is the yield stress in uniaxial tension. The invariants can be expressed in coordinates [x, y] of the deviatoric plane as: > `J'`[2]:=(x^2+y^2)/2; `J'`[3]:=-(x^2-y^2/3)*y/sqrt(6);

:= J'2 + x2

2y2

2

5

:= J'3 −

⎛

⎝⎜⎜

⎞

⎠⎟⎟ − x2 y2

3y 6

6Based upon the above plastic potential we assume a quadratic yield condition > F:=k^2;

:= F k2

where k may be identified with the maximum shear stress in yielding in a state of pure shear, which may, for example, be obtained by twisting a thin cylindrical tube. Comparing with the yield stress Y in uniaxial tension, we find the following relation: > Theta:=k/Y=sqrt((1+2*alpha/9)/3);

:= Θ = kY

+ 27 6 α9

Note, the MISES criterion is characterized by alpha = 0. Then, we have:> (k/Y)[MISES]=1/sqrt(3);

= ⎛⎝⎜⎜

⎞⎠⎟⎟

kY

MISES

33

For convenience, let us introduce dimensionless coordinates in the deviatoric plane: > [xi,eta]=[x/Y,y/Y];

= [ ],ξ η⎡⎣⎢⎢

⎤⎦⎥⎥,

xY

yY

Thus, we find from the above plastic potential. > restart: > G(xi,eta)[alpha]:=-F/Y^2+`J'`[2]/Y^2+alpha*`J'`[3]/Y^3;

:= ( )G ,ξ ηα

− + + F

Y2

J'2Y2

α J'3Y3

> G(xi,eta)[alpha]:=-(27+6*alpha)/81+ (xi^2+eta^2)/2-alpha*(xi^2-eta^2/3)*eta/sqrt(6);

:= ( )G ,ξ ηα

− − + + − 13

2 α27

ξ2

2η2

2

α⎛

⎝⎜⎜

⎞

⎠⎟⎟ − ξ2 η2

3η 6

6The yield surfaces are convex for the parameter values > alpha=[-3..3/2];

= α⎡⎣⎢⎢

⎤⎦⎥⎥ .. -3

32

> for i in [-3,-1,0,1,3/2,2]do G(xi,eta)[i]:=subs(alpha=i,G(xi,eta)[alpha]) od:

> alias(th=thickness,sc=scaling,H=Heaviside,co=color):> with(plots,implicitplot):> p[1]:=implicitplot({G(xi,eta)[-1],G(xi,eta)[0],

G(xi,eta)[1],G(xi,eta)[3/2]},xi=-sqrt(2)..sqrt(2), eta=-2*sqrt(6)/3..sqrt(2.001/3),grid=[300,300],th=3, sc=constrained,co=black,

6

title="Convex and Concave Yield Loci in the Deviatoric Plane"):> p[2]:=implicitplot(G(xi,eta)[-3],xi=-1/sqrt(2)..1/sqrt(2),

eta=-1/sqrt(6)..sqrt(2/3),grid=[300,300],th=3,co=black):> p[3]:=implicitplot(G(xi,eta)[2],xi=-1.5..1.5,

eta=-2*sqrt(6)/3..1.5, axes=boxed,grid=[300,300],co=black):> p[4]:=plots[textplot]({[0.8,1.25,`alpha = 2`],[1.2,0.95,`1.5`],

[1,0.6,`1`],[0.2,0.2,`-3`]}):> p[5]:=plot(0,xi=-1.5..1.5,linestyle=4,co=black):> p[6]:=plot({-1.5*H(xi),1.5*H(xi)},xi=-0.001..0.001,

eta=-2*sqrt(6)/3..1.5,linestyle=4,co=black):> plots[display](seq(p[k],k=1..6));

> In the above Figure we see the MISES cylinder for alpha = 0 with a radius of r = sqrt(2/3) = 0.82. The next example is concerned with the following yield condition: > restart:> F:=a*J[1]+kappa*J[1]^2/3+`J'`[2]=k^2;

:= F = + + a J113

κ J12

J'2 k2

where the first invariant, i.e. the influence of the hydrostatic pressure p, has been taken into account: > restart:> sigma[ij]:=J[1]*delta[ij]/3=-p*delta[ij]; J[1]:=-3*p;

:= σij = 13

J1 δij −p δij

7

:= J1 −3 pFor uniaxial tension with the yield stress Y we arrive from the above yield condition at the relation: > k^2/Y^2=a/Y+kappa/3+1/3;

= k2

Y2 + + aY

κ3

13

and further: > restart: macro(r=rho): > G(xi,eta)[a,p]:=

-a/Y-kappa/3-1/3-3*a*p/Y^2+3*kappa*p^2/Y^2+(xi^2+eta^2)/2;

:= ( )G ,ξ η,a p

− − − − + + + aY

κ3

13

3 a p

Y2

3 κ p2

Y2

ξ2

2η2

2> G(xi,eta)[a,p]:=0; a/Y=A; p/Y=P; xi^2+eta^2=r^2;

:= ( )G ,ξ η,a p

0

= aY

A

= pY

P

= + ξ2 η2 ρ2

> r(P,kappa,A):=sqrt(2/3+2*A+2*kappa/3+6*A*P-6*kappa*P^2);

:= ( )ρ , ,P κ A + + + − 6 18 A 6 κ 54 A P 54 κ P2

3> # example: A = 1/5 > r(P,kappa,1/5):=subs(A=1/5,r(P,kappa,A));

:= ⎛⎝⎜⎜

⎞⎠⎟⎟ρ , ,P κ

15

+ + − 485

6 κ545

P 54 κ P2

3> for i in [-1/2,0,1/4,1/2] do

r(P,i,1/5):=subs(kappa=i,r(P,kappa,1/5)) od:> with(plots):> alias(sc=scaling,H=Heaviside,th=thickness,co=color):> p[1]:=plot({-sqrt(2/3),sqrt(2/3),r(P,-1/2,1/5),r(P,0,1/5),

r(P,1/4,1/5),r(P,1/2,1/5),-r(P,-1/2,1/5),-r(P,0,1/5), -r(P,1/4,1/5),-r(P,1/2,1/5)},P=0..3/2,-1.7..1.7, sc=constrained,xtickmarks=3,ytickmarks=6,th=3,co=black, title="A = 1/5 # kappa = [-1/2, 0, 1/4, 1/2]"):

> p[2]:=plot({-1.7*H(P-1.5),1.7*H(P-1.5)},P=1.499..1.501, linestyle=4,co=black):

> p[3]:=plots[textplot]({[0.27,1.5,`-1/2`],[1.25,1.5,`0`], [0.6,0.5,`1/2`],[1.13,0.5,`1/4`], [2.1,0.82,`MISES cylinder`]}):

> plots[display](seq(p[k],k=1..3));

8

In this Figure we see three convex rotationally symmetrical yield loci with the parameters kappa = [0, 1/4, 1/2] and one concave yield surface with kappa = -1/2. In addition the MISES cylinder is represented, which has a radius of rho = sqrt(2/3) = 0.82. For P := p/Y = 0 we arrive at the deviatotic plane, where the yield loci are circles with radii depending on kappa: > r(0,kappa,1/5):=subs(P=0,r(P,kappa,1/5));

:= ⎛⎝⎜⎜

⎞⎠⎟⎟ρ , ,0 κ

15

+ 485

6 κ

3> for i in [-1/2,0,1/4,1/2] do

r(0,i,1/5):=evalf(subs(kappa=i,r(0,kappa,1/5)),4) od:> > r(0,kappa,1/5)[kappa=[-1/2,0,1/4,1/2]]=[0.86,1.03,1.11,1.18];

=

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

+ 485

6 κ

3 = κ [ ], , , / -1 2 0 / 1 4 / 1 2

[ ], , ,0.86 1.03 1.11 1.18

The parameter A is responsible for the strength differential effect, which has not been taken into account (A = 0) in the following example: > # example: A = 0 > r(P,kappa,0):=subs(A=0,r(P,kappa,A));

:= ( )ρ , ,P κ 0 + − 6 6 κ 54 κ P2

3> for i in [-1/4,-1/8,0,1/8,1/4] do

9

r(P,i,0):=subs(kappa=i,r(P,kappa,0)) od:> p[1]:=plot({r(P,-1/4,0),r(P,-1/8,0),r(P,0,0),r(P,1/8,0),

r(P,1/4,0),-r(P,-1/4,0),-r(P,-1/8,0),-r(P,0,0), -r(P,1/8,0),-r(P,1/4,0)},P=0..1.2,-1.2..1.2, th=3,co=black,xtickmarks=3,ytickmarks=4,sc=constrained, title="A = 0 # kappa = [ -1/4, -1/8, 0, 1/8, 1/4]"):

> p[2]:=plot({-sqrt(2/3)*H(P-1.2),sqrt(2/3)*H(P-1.2)}, P=1.199..1.201,linestyle=4,co=black):

> p[3]:=plots[textplot]({[0.5,1.1,`-1/4`],[1.1,1.1,`-1/8`], [0.95,0.5,`1/8`],[0.5,0.5,`1/4`], [1.55,0.82,`MISES cylinder`]},co=black):

> plots[display](seq(p[k],k=1..3));

> > r(0,kappa,0):=subs(P=0,r(P,kappa,0));

:= ( )ρ , ,0 κ 0 + 6 6 κ3

> for i in [-1/4,-1/8,0,1/8,1/4] do r(0,i,0):=evalf(subs(kappa=i,r(0,kappa,0)),3) od:

> r(0,kappa,0)[kappa=[-1/4,-1/8,0,1/8,1/4]]= [0.71,0.76,0.82,0.87,0.91];

= ⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ 6 6 κ3

= κ [ ], , , , / -1 4 / -1 8 0 / 1 8 / 1 4

[ ], , , ,0.71 0.76 0.82 0.87 0.91

> In the next Figure the parameter kappa has been assumed to be constant, while the parameter A has

10

been varied: > # example: kappa = 1/4, A = [-1/5,0,1/5] > r(P,1/4,A):=subs(kappa=1/4,r(P,kappa,A));

:= ⎛⎝⎜⎜

⎞⎠⎟⎟ρ , ,P

14

A + + −

152

18 A 54 A P272

P2

3> for i in [-1/5,0,1/5] do

r(P,1/4,i):=subs(A=i,r(P,1/4,A)) od:> p[1]:=plot({sqrt(2/3),r(P,1/4,-1/5),r(P,1/4,0),r(P,1/4,1/5),

-sqrt(2/3),-r(P,1/4,-1/5),-r(P,1/4,0), -r(P,1/4,1/5)}, P=0..1.5,-1.5..1.5,sc=constrained,th=3,co=black,xtickmarks=3):

> p[2]:=plot({-sqrt(2/3)*H(P-1.5),sqrt(2/3)*H(P-1.5)}, P=1.499..1.501,linestyle=4,co=black, title="kappa = 1/4 # A = [-1/5, 0, 1/5]"):

> p[3]:=plots[textplot]({[0.32,0.4,`-1/5`],[0.76,0.4,`0`], [1,1.15,`1/5`],[2,0.82,`MISES cylinder`]},co=black):

> plots[display](seq(p[k],k=1..3));

>

Parameter Identification

The two parameters a and k in the quadratic yield condition > restart:> F:=a*J[1]+kappa*J[1]^2/3+`J'`[2]=k^2;

:= F = + + a J113

κ J12

J'2 k2

represented in the foregoing section should be expressed in terms of Y[t] and Y[c], the yield stresses in uniaxial tension and compression, respectively. Thus, we carry out two experiments:

11

> # uniaxial tension test: > [sigma[I]=Y[t],sigma[II]=0,sigma[III]=0];

[ ], , = σI Yt = σII 0 = σIII 0> a*Y[t]+(1+kappa)*Y[t]^2/3=k^2;

= + a Yt13

( ) + 1 κ Yt2

k2

> # uniaxial compression test:> [sigma[I]=-Y[c],sigma[II]=0,sigma[III]=0];

[ ], , = σI −Yc = σII 0 = σIII 0> -a*Y[c]+(1+kappa)*Y[c]^2/3=k^2;

= − + a Yc13

( ) + 1 κ Yc2

k2

Solving these two equations, we find: > restart:> a:=(1+kappa)*(Y[c]-Y[t])/3; k^2=(1+kappa)*Y[c]*Y[t]/3;

:= a13

( ) + 1 κ ( ) − Yc Yt

= k2 13

( ) + 1 κ Yc Yt

> F:=(1+kappa)*(Y[c]-Y[t])*J[1]/3+kappa*J[1]^2+ `J'`[2]=(1+kappa)*Y[c]*Y[t]/3;

:= F = + + 13

( ) + 1 κ ( ) − Yc Yt J1 κ J12

J'213

( ) + 1 κ Yc Yt

Comparing with the parameters of the foregoing session, we find: > A:=a/Y[t]=(1+kappa)*(Q-1)/3; k^2/Y[t]^2=(1+kappa)*Q/3;

:= A = 13

( ) + 1 κ ( ) − Yc Yt

Yt

( ) + 1 κ ( ) − Q 13

= k2

Yt2

( ) + 1 κ Q3

where Q is the quotient of the yield stresses in compression and tension: > restart: > Q:=Y[c]/Y[t];

:= QYc

Yt

Furthermore, we introduce the geometrical mean value as a refernce equivalent stress:> restart: > sigma[eq]:=sqrt(Y[c]*Y[t]);

:= σeq Yc Yt

> [`x'`,`y'`]=[x/sigma[eq],y/sigma[eq]];

12

= [ ],x' y'⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥,

x

Yc Yt

y

Yc Yt

> # dimensionless hydrostatic pressure> `P'`:=p/sigma[eq];

:= P'p

Yc Yt

> rho:=sqrt(`x'`^2+`y'`^2);

:= ρ + x'2 y'2

With these abbreviations we arrive at the following representation of the above yield condition: > restart: macro(r=rho,ka=kappa):> r(`P'`,ka,Q):=sqrt(2*(1+ka)/3+

2*(1+ka)*((Q-1)/sqrt(Q))*`P'`-6*ka*`P'`^2);

:= ( )ρ , ,P' κ Q

+ + − 6 6 κ18 ( ) + 1 κ ( ) − Q 1 P'

Q54 κ P'2

3> # example: Q=6/5> r(`P'`,ka,6/5):=subs(Q=6/5,%);

:= ⎛⎝⎜⎜

⎞⎠⎟⎟ρ , ,P' κ

65

+ + − 6 6 κ3 ( ) + 1 κ 6 5 P'

554 κ P'2

3> for i in [-1/2,0,1/4,1/2] do

r(`P'`,i,6/5):=subs(ka=i,r(`P'`,ka,6/5)) od:> with(plots):> alias(sc=scaling,H=Heaviside,th=thickness,co=color):> p[1]:=plot({-sqrt(2/3),sqrt(2/3),r(`P'`,-1/2,6/5),

r(`P'`,0,6/5),r(`P'`,1/4,6/5),r(`P'`,1/2,6/5), -r(`P'`,-1/2,6/5),-r(`P'`,0,6/5),-r(`P'`,1/4,6/5), -r(`P'`,1/2,6/5)},`P'`=0..1.5,-1.5..1.5, sc=constrained,xtickmarks=3,th=3,co=black):

> p[2]:=plot({-1.1*H(`P'`-1.5),1.1*H(`P'`-1.5)}, `P'`=1.499..1.501,linestyle=4,co=black, title="Q = 6/5 # kappa = [-1/2, 0, 1/4, 1/2]"):

> p[3]:=plots[textplot]({[0.35,1.25,`-1/2`],[1.2,1.25,`0`], [1,0.3,`1/4`],[0.5,0.3,`1/2`], [2,0.81,`MISES cylinder`]},co=black):

> plots[display](seq(p[k],k=1..3));

13

>

Yield Condition of Grade Six

The following example is concerned with the representation of a yield condition of grade six in the deviatoric plane and for plane stress. Assuming plastic incompressibility and isotropy the plastic potential has the form > restart:> F:=`J'`[2]^3+alpha*`J'`[3]^2;

:= F + J'23

α J'32

where the two invariants can be expressed in terms of the coordinates [x, y] of the deviatoric plane: > `J'`[2]:=(x^2+y^2)/2; `J'`[3]:=-(x^2-y^2/3)*y/sqrt(6);

:= J'2 + x2

2y2

2

:= J'3 −

⎛

⎝⎜⎜

⎞

⎠⎟⎟ − x2 y2

3y 6

6Assuming the yield condition F = k^6 and introducing the uniaxial yield stress Y as a reference stress, we find: > F:=k^6; (k/Y)^6=(1/27)*(1+4*alpha/27);

:= F k6

= k6

Y6 + 127

4 α729

14

The invariants should be expressed in terms of dimensionsless coordinates: > [xi,eta]=[x/Y,y/Y];

= [ ],ξ η⎡⎣⎢⎢

⎤⎦⎥⎥,

xY

yY

> restart:> `J'`[2]/Y^2=(xi^2+eta^2)/2;

`J'`[3]/Y^3=-(xi^2-eta^2/3)*eta/sqrt(6);

= J'2Y2 +

ξ2

2η2

2

= J'3Y3 −

⎛

⎝⎜⎜

⎞

⎠⎟⎟ − ξ2 η2

3η 6

6Thus, the yield criterion can be expressed as: > G(xi,eta)[alpha]:=-(k/Y)^6+(`J'`[2]/Y^2)^3+

alpha*(`J'`[3]/Y^3)^2;

:= ( )G ,ξ ηα

− + + k6

Y6

J'23

Y6

α J'32

Y6

> G(xi,eta)[alpha]:=-1/27-4*alpha/729+(xi^2/2+eta^2/2)^3+ alpha*((xi^2-eta^2/3)*eta/sqrt(6))^2;

:= ( )G ,ξ ηα

− − + + 127

4 α729

⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

ξ2

2η2

2

3 α⎛

⎝⎜⎜

⎞

⎠⎟⎟ − ξ2 η2

3

2

η2

6> alpha=[-9/4, 27/8]; # limits of the convexity

= α⎡⎣⎢⎢

⎤⎦⎥⎥,

-94

278

> for i in [-5,-9/4,0,27/8,12] do G(xi,eta)[i]:=subs(alpha=i,G(xi,eta)[alpha]) od:

> alias(th=thickness,sc=scaling,co=color):> with(plots,implicitplot):> p[1]:=implicitplot({G(xi,eta)[-9/4],G(xi,eta)[0],

G(xi,eta)[27/8]},xi=-1..1,eta=-1..1,ytickmarks=6, grid=[300,300],sc=constrained,th=3,co=black):

> p[2]:=implicitplot({G(xi,eta)[-5],G(xi,eta)[12]}, xi=-1..1,eta=-1.5..1.5,linestyle=3,grid=[300,300],co=black, title="Convex and Concave Yield Loci on the Deviatoric Plane"):

> plots[display](p[1],p[2]);

15

In this Figure we see three convex (alpha = [-9/4, 0, 27/8]) and two concave [-5, 12] yield loci. The MISES circle is characterized by the parameter alpha = 0. In the next Figure the above yield condition is represented in the principal stress plane.> restart: macro(sig=sigma):> dimensionless_principal_stresses:=

[S[I]..S[III]]=[sig[I]/Y..sig[III]/Y];

:= dimensionless_principal_stresses = [ ] .. SI SIII

⎡

⎣⎢⎢

⎤

⎦⎥⎥ ..

σI

Y

σIII

YAssuming plane stress (sigma[III] = 0), the deviatoric invariants can be expressed as: > restart:> `J'`[2]:=Y^2*(S[I]^2+S[II]^2-S[I]*S[II])/3;

:= J'213

Y2 ( ) + − SI2

SII2

SI SII

> `J'`[3]:=2*Y^3*(S[I]^3+S[II]^3-3*(S[I]+S[II])*S[I]*S[II]/2)/27;

:= J'3227

Y3 ⎛⎝⎜⎜

⎞⎠⎟⎟ + − SI

3SII

3 32

( ) + SI SII SI SII

In a similar way as before we arrive from the plastic potential F = k^2 at the relation > G(S[I],S[II])[alpha]:=-1-4*alpha/27+

(S[I]^2+S[II]^2-S[I]*S[II])^3+ 4*alpha*(S[I]^3+S[II]^3-3*(S[I]+S[II])*S[I]*S[II]/2)^2/27;

:= ( )G ,SI SIIα

− − + + 14 α27

( ) + − SI2

SII2

SI SII

3 427

α⎛⎝⎜⎜

⎞⎠⎟⎟ + − SI

3SII

3 32

( ) + SI SII SI SII

2

> convexity_range:=alpha=[-9/4...27/8];

:= convexity_range = α⎡⎣⎢⎢

⎤⎦⎥⎥ ..

-94

278

> for i in [-5,-9/4,0,27/8,12] do G(S[I],S[II])[i]:=subs(alpha=i,G(S[I],S[II])[alpha]) od:

> alias(th=thickness,sc=scaling,H=Heaviside,co=color):

16

> with(plots,implicitplot):> p[1]:=implicitplot({G(S[I],S[II])[-9/4],G(S[I],S[II])[0],

G(S[I],S[II])[27/8]},S[I]=-1.5..1.5,S[II]=-1.5..1.5, grid=[300,300],sc=constrained,th=3,co=black):

> p[2]:=implicitplot({S[II]-S[I],S[II]+S[I],G(S[I],S[II])[-5], G(S[I],S[II])[12]},S[I]=-1.5..1.5,S[II]=-1.5..1.5, axes=boxed,linestyle=4,grid=[300,300],co=black, title="Convex and Concave Yield Loci for Plane Stress"):

> p[3]:=plot(0,S[I]=-1.5..1.5,co=black):> p[4]:=plot({-1.5*H(S[I]),1.5*H(S[I])},

S[I]=-0.001..0.001,S[II]=-1.5..1.5,co=black):> plots[display](seq(p[k],k=1..4));

In this Figure there are three convex (alpha = [-9/4, 0, 27/8]) and two concave (alpha = [-5, 12]) yield loci. For alpha = 0 we arrive at the MISES ellipse. Similarly, let's represent the quadratic yield criterion including the first invariant: > restart: macro(ka=kappa,sig=sigma):> # plastic potential> F:=a*J[1]+ka*J[1]^2/3+`J'`[2]=k^2;

:= F = + + a J113

κ J12

J'2 k2

> # principal stresses in a state of plane stress> plane_stress:=[sig[I]/Y,sig[II]/Y,0]=[S[I],S[II],0];

:= plane_stress = ⎡

⎣⎢⎢

⎤

⎦⎥⎥, ,

σI

Y

σII

Y0 [ ], ,SI SII 0

> J[1]:=Y*(S[I]+S[II]);

:= J1 Y ( ) + SI SII

> `J'`[2]:=Y^2*(S[I]^2+S[II]^2-S[I]*S[II])/3;

:= J'213

Y2 ( ) + − SI2

SII2

SI SII

> # uniaxial tension

17

> [sig[I],sig[II]]=[Y,0]; [S[I],S[II]]=[1,0];

= [ ],σI σII [ ],Y 0

= [ ],SI SII [ ],1 0> k^2/Y^2=A+(1+ka)/3; # A:=a/Y

= k2

Y2 + + A13

κ3

> G(S[I],S[II],A)[ka]:=-A-(1+ka)/3+A*(S[I]+S[II])+ ka*(S[I]+S[II])^2/3+(S[I]^2+S[II]^2-S[I]*S[II])/3;

:= ( )G , ,SI SII Aκ

− − − + + + + − A13

κ3

A ( ) + SI SII13

κ ( ) + SI SII2 1

3SI

2 13

SII2 1

3SI SII

> # example: A = 1/5> G(S[I],S[II],1/5)[ka]:=subs(A=1/5,%);

:= ⎛⎝⎜⎜

⎞⎠⎟⎟G , ,SI SII

15

κ

− − + + + + + − 815

κ3

15

SI15

SII13

κ ( ) + SI SII2 1

3SI

2 13

SII2 1

3SI SII

> for i in [-1/2,0,1/4,1/2] do G(S[I],S[II],1/5)[i]:=subs(ka=i,G(S[I],S[II],1/5)[ka]) od:

> alias(sc=scaling,th=thickness,H=Heaviside,co=color):> with(plots,implicitplot):> p[1]:=implicitplot({G(S[I],S[II],1/5)[0],

G(S[I],S[II],1/5)[1/4],G(S[I],S[II],1/5)[1/2]}, S[I]=-3..1.5,S[II]=-3..1.5,grid=[300,300], th=3,sc=constrained,co=black, title="A =1/5 # kappa = [-1/2, 0, 1/4, 1/2]"):

> p[2]:=implicitplot(G(S[I],S[II],1/5)[-1/2], S[I]=-3..1.5,S[II]=-3..1.5,grid=[300,300],co=black):

> p[3]:=implicitplot(S[II]-S[I],S[I]=-2.2..1, S[II]=-2.2..1,linestyle=4,co=black):

> p[4]:=implicitplot(S[II]+S[I],S[I]=-1..1, S[II]=-1..1,linestyle=4,co=black):

> p[5]:=plots[textplot]({[-1.2,-2.5,`kappa=`], [-0.75,-2.3,`0`],[-0.75,-1.8,`1/4`],[-0.75,-1.1,`1/2`]}):

> plots[display](seq(p[k],k=1..5));

18

In this Figure we see three cnvex yield loci for positive kappa values [0, 1/4, 1/2] and one concave curve for the negative value kappa = -1/2. Because of the assumption of isotropy the yield loci are symmetric with respect to the 45°-line. The parameter A characterizes the strength differential effect as has been discussed in the chapter parameter identification: > restart:> A:=(1+kappa)*(Q-1)/3;

:= A( ) + 1 κ ( ) − Q 1

3where Q is the quotient of the yield stresses in compression and tension:> restart:> Q:=Y[c]/Y[t]=1+3*A/(1+kappa);

:= Q = Yc

Yt + 1

3 A + 1 κ

For A = 1/5 and kappa = [0, 1/4, 1/2] we find the strength differential effect:> restart:> for i in [0,1/4,1/2] do Q(A=1/5)[kappa=i]:=

evalf(subs({A=1/5,kappa=i},1+3*A/(1+kappa)),3) od;

:= ⎛⎝⎜⎜

⎞⎠⎟⎟Q = A

15

= κ 0

1.60

:= ⎛⎝⎜⎜

⎞⎠⎟⎟Q = A

15

= κ / 1 4

1.48

:= ⎛⎝⎜⎜

⎞⎠⎟⎟Q = A

15

= κ / 1 2

1.40

One can read these values on the abscissa in the above Figure.

19

> > restart: macro(ka=kappa):> G(S[I],S[II],A)[ka]:=-A-(1+ka)/3+A*(S[I]+S[II])+

ka*(S[I]+S[II])^2/3+(S[I]^2+S[II]^2-S[I]*S[II])/3;

:= ( )G , ,SI SII Aκ

− − − + + + + − A13

κ3

A ( ) + SI SII13

κ ( ) + SI SII2 1

3SI

2 13

SII2 1

3SI SII

> # example: A = 0> G(S[I],S[II],0)[ka]:=subs(A=0,%);

:= ( )G , ,SI SII 0κ

− − + + + − 13

κ3

13

κ ( ) + SI SII2 1

3SI

2 13

SII2 1

3SI SII

> for i in [-1/2,0,1/4,1/2] do G(S[I],S[II],0)[i]:=subs(ka=i,G(S[I],S[II],0)[ka]) od:

> alias(sc=scaling,th=thickness,H=Heaviside,co=color):> with(plots,implicitplot):> p[1]:=implicitplot({G(S[I],S[II],0)[0],

G(S[I],S[II],0)[1/4],G(S[I],S[II],0)[1/4], G(S[I],S[II],0)[1/2]},S[I]=-1.5..1.5,S[II]=-1.5..1.5, sc=constrained,grid=[300,300],th=3,co=black, title="A = 0 # [-1/2, 0, 1/4, 1/2]"):

> p[2]:=implicitplot(G(S[I],S[II],0)[-1/2], S[I]=-1.5..1.5,S[II]=-1.5..1.5,axes=boxed, grid=[300,300],co=black):

> p[3]:=implicitplot(S[II]-S[I],S[I]=-1.1..1.1, S[II]=-1.1..1.1,linestyle=4,co=black):

> p[4]:=implicitplot(S[II]+S[I],S[I]=-0.8..0.8, S[II]=-0.8..0.8,linestyle=4,co=black):

> p[5]:=plot(0,S[I]=-1.5..1.5,S[II]=-1.5..1.5, co=black):> p[6]:=plot({-1.5*H(S[I]),1.5*H(S[I])},

S[I]=-0.001..0.001,co=black):> p[7]:=plots[textplot]({[-1.1,-1.35,`kappa=`],

[-0.8,-1.25,`0`],[-0.8,-0.95,`1/4`],[-0.8,-0.4,`1/2`]}):> plots[display](seq(p[k],k=1..7));

20

In this Figure there are three convex yield loci for kappa = [0, 1/4, 1/2] and one concave curve for kappa = -1/2. Because of A = 0 there is no strength differential effect. For kappa = 0 we arrive at the MISES ellipse. In earlier investigations other aspects concernig isotropic and incompressible or compressible plastic yielding have been taken in consideration by BERTRAM (2005), BETTEN (1975, 2001), LEMAITRE & CHABOCHE (1990), SKRZYPEK & HETNARSKI (1993), to name just a few.

Orthotropic Behaviour

The above assumption of isotropy relates to ananalytical approximate description of the behaviour of a material, which leads (especially, when large plastic deformations are considered) to inexact results. Primarily this is explained by the directional dependence of the mechanical properties. This phenomenon is called mechanical anisotropy. In rare circumstances the properties are completely irregular with respect to the orientation. Usually we can specify planes of symmetry, with respect of which the changes are symmetric. If there are three mutually perpendicular symmetric planes, the material is called orthotropic. The following yield criterion (HILL-condition) is often used for rolled steel: > restart: macro(sig=sigma):> H[I]*(sig[II]-sig[III])^2+H[II]*(sig[III]-sig[I])^2+

H[III]*(sig[I]-sig[II])^2=1;

= + + HI ( ) − σII σIII2

HII ( ) − σIII σI2

HIII ( ) − σI σII2

1

where [I, II, III] are the principal directions of orthotropy. This condition is not valid for plastic compressible materials, since a hydrostatic stress state (sigma[I] = sgma[II] = sigma[III]) cannot satisfy this condition. For deep drawing sheets the material should behave isotropic in the sheet

21

plane in order to avoid the development of scallops. Such behaviour is called to be transversely isotropic. Then, the HILL-condition is symmetric with respect to the 45°-line. Thus, we have: > H[I]=H[II]; H[III]/H[I]=R; 1/(H[I]+H[III])=Y^2;

= HI HII

= HIII

HIR

= 1

+ HI HIIIY2

> sig[I]^2+sig[II]^2-2*R*sig[I]*sig[II]/(1+R)=Y^2;

= + − σI2

σII2 2 R σI σII

+ 1 RY2

> G(S[I],S[II],R):=-1+S[I]^2+S[II]^2-2*R*S[I]*S[II]/(1+R);

:= ( )G , ,SI SII R − + + − 1 SI2

SII2 2 R SI SII

+ 1 R> > for i in [-2,0,1/4,1] do

G(S[I],S[II],i):=subs(R=i,G(S[I],S[II],R)) od:> alias(th=thickness,sc=scaling,H=Heaviside,co=color):> with(plots,implicitplot): > p[1]:=implicitplot({G(S[I],S[II],0),G(S[I],S[II],1/4),

G(S[I],S[II],1)},S[I]=-1.5..1.5,S[II]=-1.5..1.5, sc=constrained,axes=boxed,grid=[300,300],th=3,co=black):

> p[2]:=implicitplot(G(S[I],S[II],-2),S[I]=-1.5..1.5, S[II]=-1.5..1.5,grid=[300,300],co=black):

> p[3]:=implicitplot({S[I]-S[II],S[I]+S[II]}, S[I]=-1.25..1.25,S[II]=-1.25..1.25,linestyle=4,co=black):

> p[4]:=plot({-1.5*H(S[I]),1.5*H(S[I])},S[I]=-0.01..0.01, S[II]=-1.5..1.5,linestyle=4,co=black):

> p[5]:=plot(0,S[I]=-1.5..1.5,linestyle=4,co=black, title="Transverse Isotropy # Parameter R = [0, 1/4, 1, -2]"):

> p[6]:=plots[textplot]({[0.42,1.35,`R = -2`], [0.38,0.75,`R = 0`],[1,1.25,`R = 1`]}):

> plots[display](seq(p[k],k=1..6));

22

In this Figure we see three convex yield loci for R = [0, 1/4, 1] and one concave curve for R = -2. For R = 1 we arrive at the MISES ellipse. The last two Figures show identical curves although the corresponding parameters have different mechanical meanings. The first Figure is valid for isotropic and compressible materials (parameter kappa), while the parameter R in the second Figure characterizes the anisotropic property of the material with a single preferred direction at every point called to be transversely isotropic. We have selected the parameter R = (1 - 2*kappa) / (1+ 4*kappa) .Thus, the two figures are congruent. For practical use we have to carry out suitable experimental tests in order to determine the parameters kappa and R depending on the given material. The yield loci shown in the last Figure can be represented in the deviatoric plane as follow: > restart: macro(sig=sigma):> G(S[I],S[II],R):=-1+S[I]^2+S[II]^2-2*R*S[I]*S[II]/(1+R);

:= ( )G , ,SI SII R − + + − 1 SI2

SII2 2 R SI SII

+ 1 R> restart:> # transformation on the deviatoric plane > S[I]:=-(xi+sqrt(3)*eta)/sqrt(2);

S[II]:=(xi-sqrt(3)*eta)/sqrt(2);

:= SI −( ) + ξ 3 η 2

2

23

:= SII( ) − ξ 3 η 2

2> G(xi,eta,R):=simplify(-1+S[I]^2+S[II]^2-2*R*S[I]*S[II]/(1+R));

:= ( )G , ,ξ η R− − + + + 1 R ξ2 2 ξ2 R 3 η2

+ 1 R> `G'`(xi,eta,R):=-1+(xi/A)^2+(eta/B)^2;

:= ( )G' , ,ξ η R − + + 1ξ2

A2

η2

B2

> [A,B]=[sqrt((1+R)/(1+2*R)),sqrt((1+R)/3)];

= [ ],A B⎡

⎣⎢⎢

⎤

⎦⎥⎥,

+ 1 R + 1 2 R

+ 3 3 R3

Isotropy is characterized by R = 1. Then, we arrive at the MISES circle with a radius A = B = sqrt( 2/3) = 0.8165. > for i in [-2,0,1/4,1] do

G(xi,eta,i):=subs(R=i,G(xi,eta,R)) od:> alias(th=thickness,sc=scaling,co=color):> with(plots,implicitplot):> p[1]:=implicitplot({G(xi,eta,0),G(xi,eta,1/4),G(xi,eta,1)},

xi=-1..1,eta=-1..1,grid=[300,300],sc=constrained,th=3,co=black):> p[2]:=implicitplot(G(xi,eta,-2),xi=-1..1,eta=-1..1,

grid=[300,300],ytickmarks=3,co=black, title="Tranverse Isotropic # R = [-2, 0, 1/4, 1]"):

> p[3]:=plots[textplot]({[0.25,0.85,`R = 1`],[0.85,0.85,`R = -2`], [0.25,0.45,`R = 0`]}):

> plots[display](seq(p[k],k=1..3));

The parameters R = [0, 1/4] furnish two ellipses, R = 1 expresses the MISES circle,

24

and R = -2 generates a hyperbola.

References

BERTRAM, A. (2005). Elasticity and Plasticity of Large Deformation, Springer-Verlag, Berlin / Heidelberg / New York.

BETTEN, J. (1975). Beitrag zum isotropen kompressiblen plastischen Fließen, Archiv Eisenhüttenwesen, 46: 317 - 323.

BETTEN, J. (1982). Integrity basis for a second-order and a fourth-order tensor, Int. J. Math. Math. Sci. 5: 87 - 96.

BETTEN, J. (1985). The classical plastic potential theory in comparison with the tensor function theory, Eng. Fracture Mech. 21: 641 - 652. Presented at the Int. Symposium Plasticity Today, Udine, June 1983.

BETTEN, J. (1987). Tensorrechnung für Ingenieure, Teubner-Verlag, Stuttgart.

BETTEN, J. (1988). Applications of Tensor Functions to the Formulation of Yield Criteria for Anisotropic Materials, Int. J. of Plasticity 4: 29 - 46.

BETTEN, J. (1998). Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien, ZAMM 78: 507 - 521. Hauptvortrag auf der 75. GAMM-Tagung in Regensburg, März 1997.

BETTEN, J. (2001). Kontinuumsmechanik, 2. Aufl., Springer-Verlag, Berlin / Heidelberg.

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