Algorithmic tangent modulus at finite strains based on multiplicative decomposition

Appl. Math. Mech. -Engl. Ed., 35(3), 345–358 (2014)DOI 10.1007/s10483-014-1795-6c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2014

Applied Mathematicsand Mechanics(English Edition)

Algorithmic tangent modulus at finite strains based onmultiplicative decomposition∗

Chao-jun LI (��)1,2, Ji-li FENG (��)1,2

(1. State Key Laboratory for Geomechanics and Deep Underground Engineering,

Beijing 100083, P. R. China;

2. School of Mechanics and Civil Engineering, China University of Mining and

Technology (Beijing), Beijing 100083, P. R. China)

Abstract The algorithmic tangent modulus at finite strains in current configuration

plays an important role in the nonlinear finite element method. In this work, the exact

tensorial forms of the algorithmic tangent modulus at finite strains are derived in the

principal space and their corresponding matrix expressions are also presented. The al-

gorithmic tangent modulus consists of two terms. The first term depends on a specific

yield surface, while the second term is independent of the specific yield surface. The

elastoplastic matrix in the principal space associated with the specific yield surface is

derived by the logarithmic strains in terms of the local multiplicative decomposition. The

Drucker-Prager yield function of elastoplastic material is used as a numerical example to

verify the present algorithmic tangent modulus at finite strains.

Key words algorithmic tangent modulus, matrix expression, finite strain, multiplica-

tive decomposition

Chinese Library Classification O344.1

2010 Mathematics Subject Classification 74B20

1 Introduction

The initial developments of elastoplastic constitutive relation at finite strains depend onthe use of hypoelastic-based constitutive formulations which are characterized in terms of suit-ably chosen objective stress rates and extended from standard infinitesimal elastoplasticitymodels[1–2]. The hypoelastic-based descriptions have two controversial issues which includefundamental drawbacks such as the possible lack of the objectivity of (algorithmic) incremen-tal constitutive laws[3] and dissipative behaviors within the “elastic” range[4]. In the above-mentioned context, the hyperelastic-based formulations of finite plasticity by the local mul-tiplicative decomposition have emerged[5], which naturally bypass the inherent drawbacks ofhypoelastic-based approaches. As a consequence, the dissipative response becomes impossiblewithin the elastic range. Another merit is that the requirement of the incremental objectiv-ity (the frame invariance of the algorithmic constitutive rule) is trivially satisfied. Generally

∗ Received Jan. 9, 2013 / Revised May 31, 2013Project supported by the National Natural Science Foundation of China (Nos. 41172116, U1261212,and 51134005)Corresponding author Ji-li FENG, Professor, Ph.D., E-mail: [email protected]

346 Chao-jun LI and Ji-li FENG

speaking, the closest point projection algorithm is applied for the analysis of the elastoplasticconstitutive equation. The algorithm is based on the elastic predictor-plastic corrector methodwhich deals with two steps. The first step is to compute the trial stress with no plastic strain,i.e., all strains are considered as elastic strains, and the second step is to analyze the yieldfunction value to check the yield condition according to the trial stress. If the elastic stateis violated, the trial stress must be returned to the yield surface according to the discreteKuhn-Tucker condition. Otherwise, the trial stress is the real stress and exits. In the nonlinearfinite element method, the algorithmic modulus must be computed and the chosen algorithmictangent modulus can gain the quadratic convergence rate when using the Newton-Raphsonmethod. Consequently, the analysis of the algorithmic tangent modulus in the current configu-ration plays an important role in the nonlinear finite element method. Though the algorithmictangent modulus is well understood under small deformations in the literature, there are someambiguities when considering finite strains. The algorithmic tangent modulus in the principalspace is one valid method to investigate the finite deformation of solid. The derivations ofthe algorithmic tangent modulus at finite strains in the principal space have been presentedfor the case λ1 �= λ2 �= λ3 and the concise forms have been obtained[6]. However, the matrixexpressions what we expect in computational mechanics are not obtained. In particular, theother two cases that include λ1 = λ2 �= λ3 and λ1 = λ2 = λ3 are not clear, and their matrixexpressions are also not provided. Moreover, the valid matrix expressions of the fourth-ordertensor are controversial issues[7].

This paper is organized as follows. In Section 2, the exact tensorial forms of the algorith-mic tangent modulus at finite strains are derived, and their corresponding matrix expressionsare given in Section 3. The elastoplastic matrix in the principal space associated with thespecific yield surface is derived by the logarithmic strain in terms of the local multiplicativedecomposition in Section 4. The Drucker-Prager yield function of elastoplastic material is usedas a numerical example to verify the present algorithmic tangent modulus at finite strains inSection 5. In the present work, the number of the underlines of a notation corresponds to itsorder, e.g., C represents a first-order tensor, C is a second-order tensor, and C indicates a

fourth-order tensor; and the symbol “:” in equations denotes as a double contract operator oftwo tensors, e.g., b : c means a scalar, and b : d represents a second-order tensor.

2 Tensorial forms of algorithmic tangent modulus at finite strains

The local multiplicative decomposition of the deformation gradient F is

F = F e · F p, (1)

where F e and F p are the elastic and plastic parts of the deformation gradient, respectively[11].The right and left Cauchy-Green deformation tensors are defined, respectively, as

C = FT · F , b = F · FT. (2)

The logarithmic strain tensor is defined as

ε =12

ln b. (3)

In the principal space, the Kirchhoff stress tensor τ and the left Cauchy-Green deformationtensor b are coaxial because of the restriction of isotropy, and their respective spectral decom-positions take the forms[8]

b =3∑

A=1

λ2AnA ⊗ nA, τ =

3∑

A=1

βAnA ⊗ nA, (4)

Algorithmic tangent modulus at finite strains based on multiplicative decomposition 347

where λ2A is the eigenvalue of the left Cauchy-Green deformation tensor, βA indicates the

eigenvalue of the Kirchhoff stress, and nA (A = 1, 2, 3) represent the eigenvectors in the currentconfiguration.

The eigenvalues of the right Cauchy-Green deformation tensors C are λ2A (A = 1, 2, 3),

while their corresponding eigenvectors are NA in the reference configuration, expressed asNA = F−1 · nA. Consequently, its spectral decomposition is

C =3∑

A=1

λ2ANA ⊗ NA. (5)

Let us define

MA = λ−2A NA ⊗ NA, A = 1, 2, 3. (6)

For the case λ1 �= λ2 �= λ3, MA can also be denoted as[6]

MA =C − (I1 − λ2

A)E + I3λ−2A C−1

DA, (7)

where DA = 2λ4A − I1λ

2A + I3λ

−2A , and E is the second-order identity tensor.

In the reference configuration, the algorithmic tangent modulus is defined as

D = 2∂S

∂C, (8)

where S =3∑

A=1

βAMA is the second Piola-Kirchhoff stress tensor.

From Eq. (8), the algorithmic tangent modulus expressed in the current configuration canbe obtained as follows:

d = F · (F · D · F ) · F. (9)

The following three basic relations are given:

λ1 �= λ2 �= λ3,∂λA

∂C=

12λAMA, (10)

λ = λ1 = λ2 �= λ3,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

∂λ3

∂C=

12λ3M

3,

∂λ

∂C=

14λ(C−1 − M3),

(11)

λ = λ1 = λ2 = λ3,∂λ

∂C=

16λC−1, (12)

which are proved in Appendix A.The exact derivations of the algorithmic tangent modulus in the current configuration are

given for the three different cases as follows.Case I λ1 �= λ2 �= λ3

D = 2∂S

∂C= 2

( 3∑

A=1

∂βA

∂C⊗ MA +

3∑

A=1

βA

∂MA

∂C

). (13)


Since

∂βA

∂C=

∂βA

∂λB

∂λB

∂C=

12

∂βA

∂λBλBMB, (14)

∂MA

∂C=

1DA

(I −

(∂I1

∂C− ∂λ2

A

∂λA

∂λA

∂C

)⊗ E +

∂I3

∂Cλ−2

A ⊗ C−1

+ I3∂λ−2

A

∂λA

∂λA

∂C⊗ C−1 + I3λ

−2A

∂C−1

∂C

)

− 1D2

A

∂DA

∂C⊗ (C − (I1 − λA)E + I3λ

−2A C−1), (15)

where I1 and I3 are the first and third invariants of C, respectively.

∂I1

∂C= E,

∂I3

∂C= I3C

−1,∂λA

∂C=

12λAMA,

∂C−1

∂C= −I

C−1,

∂DA

∂C= (8λ3

A − 2I1λA − 2I3λ−3A )

12λAMA − λ2

AE + I3λ−2A C−1.

Based on Eqs. (14) and (15), Eq. (13) is exactly expressed as

D =3∑

A=1

( 3∑

B=1

aepMA ⊗ MB)

+3∑

A=1

2βACg, (16)

where

Cg =∂MA

∂C=

1DA

(I − E ⊗ E + I3λ

−2A (C−1 ⊗ C−1 − I

C−1) +

12D′

AλAMA ⊗ MA

+ λ2A(MA ⊗ E + E ⊗ MA) − I3λ

−2A (MA ⊗ C−1 + C−1 ⊗ MA)

),

aepAB =

∂βA

∂λBλB , (IC−1)ABCD =

12(C−1ACC−1BD + C−1ADC−1BC),

D′A = 8λ3

A − 2I1λA − 2I3λ−3A .

According to Eq. (9), we finally obtain

d =3∑

A=1

( 3∑

B=1

aepmA ⊗ mB)

+3∑

A=1

2βAcg, (17)

where aep derived in Section 4 is the elastoplastic modulus associated with the special yieldsurface, and cg is independent of the specific model of plasticity since the principal directions

are functions of the total deformations expressed by

cg =1

DA

(I

b− b ⊗ b + I3λ

−2A (E ⊗ E − I) − 1

2D′

AλAmA ⊗ mA

+ λ2A(mA ⊗ b + b ⊗ mA) − I3λ

−2A (mA ⊗ E + E ⊗ mA)

).


Case II λ = λ1 = λ2 �= λ3

S = βM + β3M3,

D = 2∂S

∂C=

∂β

∂C⊗ M +

∂β3

∂C⊗ M3 + β

∂M

∂C+ β3

∂M3

∂C. (18)

After some operations, the exact expression of Eq. (18) is

D =(1

4∂β

∂λλ(C−1 − M3) +

12

∂β

∂λ3λ3M

3)⊗ M

+(1

4∂β3

∂λλ(C−1 − M3) +

12

∂β3

∂λ3λ3M

3)⊗ M3

+ β(− 1

2(C−1 − M3) ⊗ M − (λ2 − λ2

3)−1

(12λ2(C−1 − M3) − λ2

3M3)⊗ M

+ λ−2(λ2 − λ23)

−1(I − λ23M

3 ⊗ E))

+ β3

(− M3 ⊗ M3 − (λ2

3 − λ2)−1(λ2

3M3 − 1

2λ2(C−1 − M3)

)⊗ M3

+ λ−23 (λ2

3 − λ2)−1(I − 1

2λ2(C−1 − M3) ⊗ E

)).

The algorithmic tangent modulus in the current configuration reads

d =(

14aep11m +

12aep13m

3

)⊗ m +

(14aep31m +

12aep33m

3

)⊗ m3

+ β(− 1

2m ⊗ m − (

λ2 − λ23

)−1(1

2λ2m − λ2

3m3)⊗ m

+ λ−2(λ2 − λ2

3

)−1(I

b− λ2

3m3 ⊗ b

))

+ β3

(− m3 ⊗ m3 − (

λ23 − λ2

)−1(λ2

3m3 − 1

2λ2m

)⊗ m3

+ λ−23

(λ2

3 − λ2)−1

(I

b− 1

2λ2m ⊗ b

)), (19)

where

aep11 =

∂β

∂λλ, aep

13 =∂β

∂λ3λ3, aep

31 =∂β3

∂λλ, aep

33 =∂β3

∂λ3λ3,

and the fourth-order tensors of the last two terms in the parentheses are independent of thespecific yield surface.

Case III λ = λ1 = λ2 = λ3

S = βM, M = λ−2E,

D = 2∂S

∂C= 2

( ∂β

∂C⊗ M + β

∂M

∂C

)= 2

(16

∂β

∂λλC−1 ⊗ M − 1

3βλ−2C−1 ⊗ E

).

The algorithmic tangent modulus in the current configuration is

d = 2(1

6aepE ⊗ E − 1

3βλ−2E ⊗ b

), (20)

where aep = ∂β∂λλ, and λ−2E ⊗ b is independent of the specific yield surface.


3 Matrix expressions of algorithmic tangent modulus in current configu-ration

As done in Ref. [7], let the second-order base-tensors be

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G1

= n1 ⊗ n1, G4

= n1 ⊗ n2 + n2 ⊗ n1,

G2

= n2 ⊗ n2, G5

= n2 ⊗ n3 + n3 ⊗ n2,

G3

= n3 ⊗ n3, G6

= n3 ⊗ n1 + n1 ⊗ n3.

(21)

Therefore, the following matrix expressions are obtained:

[E ⊗ E

]=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 0 0 01 1 1 0 0 01 1 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[b ⊗ b

]=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ41 λ2

1λ22 λ2

1λ23 0 0 0

λ22λ

21 λ4

2 λ22λ

23 0 0 0

λ23λ

21 λ2

3λ22 λ4

3 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[Ib] =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ41 0 0 0 0 00 λ4

2 0 0 0 00 0 λ4

3 0 0 00 0 0 1

2λ21λ

22 0 0

0 0 0 0 12λ2

2λ23 0

0 0 0 0 0 12λ2

1λ23

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

From the definition of Eq. (21), the matrix expressions of Eqs. (17), (19), and (20) are sum-marized in Box 1 after some operations.

Box 1 Matrix expressions of algorithmic tangent modulusCase I λ1 �= λ2 �= λ3

[d] =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

aep11 − β1 aep

12 aep13 0 0 0

aep21 aep

22 − β2 aep23 0 0 0

aep31 aep

32 aep33 − β3 0 0 0

0 0 0 β1λ22−β2λ2

1

2(λ21−λ2

2)0 0

0 0 0 0 β2λ23−β3λ2

2

2(λ22−λ2

3)0

0 0 0 0 0 β3λ21−β1λ2

3

2(λ23−λ2

1)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.


Case II λ = λ1 = λ2 �= λ3

[d] =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A11 A1214aep

31 0 0 0

A21 A2214aep

31 0 0 012aep

1312aep

1312aep

33 − β3 0 0 0

0 0 0 λ−23 λ4β3−λ2β

2(λ23−λ2) 0 0

0 0 0 0 λ2β3−λ23β

2(λ23−λ2) 0

0 0 0 0 0 λ2β3−λ23β

2(λ23−λ2)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where

A11 =λ4λ−2

3 β3 − λ23β

2 (λ23 − λ2)

+ 14aep

11, A12 =14aep11 −

λ4λ−23 β3 +

(λ2

3 − 2λ2)β

2 (λ23 − λ2)

,

A21 =14aep11 −

λ4λ−23 β3 +

(λ2

3 − 2λ2)β

2 (λ23 − λ2)

, A22 =14aep11 +

λ4λ−23 β3 − λ2

3β

2 (λ23 − λ2)

.

Case III λ = λ1 = λ2 = λ3

[d] =(1

3aep − 2

3β)

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

4 Modulus aep associated with specific yield surface

For a mixed hardening model, the general yield surface can be written as

f(σ, ξ, H) = 0, (22)

where σ = 1J τ , J , ξ, and H are the Cauchy stress tensor, the determinant of the deformation

gradient, the back stress tensor associated with the kinematic hardening, and the isotropichardening variable, respectively.

From the consistent condition, we obtain df(σ, ξ, H) = 0, i.e.,

∂f

∂σ: dσ +

∂f

∂ξ: dξ +

∂f

∂HdH = 0. (23)

The incremental Cauchy stress tensor can be denoted by the incremental logarithmic strains

dσ =1J

dτ − 1J2

τdJ =1J

(h : dεe − σ

(∂J

∂ε: dε

)), (24)

where h is the modified elastic modulus in the principal space.

If the logarithmic strain is employed, the additive decomposition of the strain is expressedas

dε = dεe + dεp. (25)


Proof Considering the isotropic hypothesis, from Eqs. (1) and (3), we have

ε =12

ln b =12

ln(F · FT) =12

ln((F e · F p) · (F pT · F eT)). (26)

After expanding Eq. (26) and recombining it, we obtain

ε = εe + εp, (27)

where εe = 12 ln(F e · F eT), and εp = 1

2 ln(F p · F pT).In the principal space, the principal directions of the logarithmic strain identify with the

left Cauchy-Green tensor, and the corresponding respective spectral decomposition is

ε =3∑

A=1

εAnA ⊗ nA, (28)

where εA = ln λA.Due to aep

AB = ∂βA

∂λBλB, we can rewrite aep

AB in the form

aepAB =

∂βA

∂εB. (29)

At finite strains, the non-associated flow rule can be, from the return mapping method, ap-proximately expressed as

dεp = dγ1J

∂gp

∂σ, (30)

where gp is the plastic potential function, which is identified by the yield surface function underthe consideration of the associated flow rule.

Combining Eqs. (23), (24), (25), and (30), after some operations, we obtain

dγ =

∂f∂σ : h −

(∂f∂σ : σ

)∂J∂ε

1J

∂f∂σ : h : ∂gp

∂σ −(

∂f∂ξ : A : ∂gp

∂σ + ∂f∂H

∂H∂εp : ∂gp

∂σ

) : dε, (31)

where A =∂ξ

∂εp is associated with the hardening modulus.

Substituting Eqs. (25) and (31) into Eq. (24), we finally obtain the exact form of the stress-strain formulation

dσ =1J

⎛

⎜⎝h − 1J

(h : ∂gp

∂σ

)⊗

(∂f∂σ : h −

(∂f∂σ : σ

)∂J∂ε

)

1J

∂f∂σ : h : ∂gp

∂σ −(

∂f∂ξ : A : ∂gp

∂σ + ∂f∂H

∂H∂εp : ∂gp

∂σ

) − σ ⊗ ∂J

∂ε

⎞

⎟⎠ = aep : dε. (32)

The corresponding matrix expression of the elastoplastic tangent modulus aep is

[aep] =1J

[h] −1

J2 [h][ ∂f∂σ ]([∂gp

∂σ ]T([h] − [σ][∂J∂ε ]T))

1J [ ∂f

∂σ ]T[h][∂gp∂σ ] − ([∂f

∂ξ ]T[A] + ∂f∂H [ ∂H

∂εp ]T)[∂gp∂σ ]

− 1J

[σ][∂J

∂ε

]T

. (33)

In Eq. (32), the term ∂J∂ε is written as[9]

∂J

∂ε=

∂J

∂b:

∂b

∂ε= 2

∂J

∂b: (L)−1, (34)


where⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

∂J

∂b=

12Jb−1, L =

∂ ln b

∂b=

(∂ea

∂a

)−1

,

a = ln b,∂ea

∂a=

∞∑

n=1

1n!

n∑

m=1

am−1 ⊗ an−m.

(35)

5 Numerical example

Large deformation of geomaterial is one of the mostly concerned issues in the mechanicalresearch field[6,12–14]. In order to examine the present algorithmic scheme, the Drucker-Pragermodel of elasto-plasticity most commonly used in the geotechnical engineering analysis with theassociated flow rule is employed in the numerical example. The model formulation with respectto three principal stresses is first presented. The algorithmic tangent modulus is then coded byMATLAB. As a comparison, the elastoplastic tangent modulus at infinitesimal strains is alsopresented.

The Drucker-Prager yield surface or function reads

f =√

J2 − αI1 − k = 0, (36)

where J2 = 16 ((σ1 − σ2)

2 + (σ3 − σ1)2 + (σ2 − σ3)

2) is the second invariant of the deviatoricstress tensor, I1 = σ1 + σ2 + σ3 is the first invariant of the stress tensor, and α and k arematerial parameters, respectively.

The first-order derivatives of the yield function with respect to the three principal stressesare

⎡

⎢⎢⎢⎢⎢⎢⎣

∂f

∂σ1

∂f

∂σ2

∂f

∂σ3

⎤

⎥⎥⎥⎥⎥⎥⎦=

⎡

⎢⎢⎢⎢⎢⎢⎣

16√

J2

(2σ1 − σ2 − σ3) − α

16√

J2

(2σ2 − σ1 − σ3) − α

16√

J2

(2σ3 − σ2 − σ1) − α

⎤

⎥⎥⎥⎥⎥⎥⎦. (37)

Under the conditions of the no hardening and associated flow rule, Eq. (33) is rewritten as

[aep] =1J

[h] − 1J

[h][ ∂f∂σ ]([ ∂f

∂σ ]T([h] − [σ][∂J∂ε ]T))

[ ∂f∂σ ]T[h][ ∂f

∂σ ]− 1

J[σ]

[∂J

∂ε

]T

. (38)

The corresponding elastoplastic tangent modulus at infinitesimal strains is simplified as follows:

[cep] = [h] − ([h][ ∂f∂σ ])([ ∂f

∂σ ]T[h])

[ ∂f∂σ ]T[h][ ∂f

∂σ ]. (39)

As a typical example of clay, Poisson’s ratio and Young’s modulus are μ = 0.28 and E =10.3MPa, respectively. The elastic modulus is expressed as

[h] =E(1 − μ)

(1 + μ)(1 − 2μ)

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 μ1−μ

μ1−μ 0 0 0

μ1−μ 1 μ

1−μ 0 0 0μ

1−μμ

1−μ 1 0 0 0

0 0 0 1−2μ1−μ 0 0

0 0 0 0 1−2μ1−μ 0

0 0 0 0 0 1−2μ1−μ

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.


Furthermore, c = 0.11MPa and φ = 0.14π are used. It is provided that the yield surface of theDrucker-Prager and Mohr-Coulomb criteria coincide at the outer edges of the Mohr-Coulombsurface. Consequently, we can obtain

α =2 sinφ√

3(3 + sin φ), k =

6c cosφ√3(3 + sin φ)

. (40)

The following procedures with one main program are developed by MATLAB to compute theelastoplastic tangent modulus of the Drucker-Prager model. It should be noted that, in Boxes2 and 3, [·]kn+1 and (·)k

n+1 are a matrix [·] and a variable at the kth iteration of the load stepduring the time [tn, tn+1], respectively.

i) Compute Eq. (34) and return a 1 × 3 matrix, and its implementation is shown in Box 2.ii) Compute the elastoplastic tangent modulus associated with the Drucker-Prager yield

surface and update the stresses and strains by means of the closest point projection method[10],which is summarized in Box 3.

iii) Compute the algorithmic tangent modulus.Box 2 Compute the derivative of J with respect to the logarithmic strains ε in the principal

space(i) Give b0. Set the initial values and the numerical tolerance ρtol: [Dea]0 = 0, n0 = 1, and

i := 1.(ii) Set [a]i = 0. Then, compute ni = ni−1 · i.(iii) For j = 1 : i, compute [a]i =

i∑j=1

bj−1 ⊗ bi−j .

(iv) Compute [Dea]i = [Dea]i−1+ [a]i

ni and Ttol = det[a]i, where Dea is a matrix correspondingto the tensor form ∂ea

∂a mentioned in Eq. (35), and Ttol is a summary of residuals at every

computational step for [Dea].Check the condition. If Ttol < ρtol, exit; else, set i := i + 1 and go to Step (ii).Box 3 Closest point projection method with no hardening and associated flow rule(i) k := 0. Assume that the total strains are elastic strains. Set the initial value of the

Lagrange multiplier as [εen+1]

0 = [εen+1]

trial (Δλ0n+1 = 0).

(ii) Compute the values of the yield function, the trial stress, and the residual as follows:

([σn+1]k)trial = [h]([εen+1]

k)trialfkn+1 = f(([σn+1]k)trial),

[Rn+1]k = [εen+1]

k − [εen+1]

trial +1J

Δλkn+1

( ∂f

∂[σn+1]

)k

.

Check the yield condition. If fkn+1 < ρtol, then set (∗)n+1 = (∗)k

n+1 and exit; else, do thefollowing calculations.

(iii) Compute the matrix [B],

[B]−1 = [h]−1 + Δλkn+1

∂2f

∂([σn+1]k)2.

(iv) Compute the incremental value of the Lagrange multiplier,

δ(Δλn+1)k =fk

n+1 −(

∂f∂[σn+1]

)k[B][Rn+1]k(

∂f∂[σn+1]

)k[B](

∂f∂[σn+1]

)k.


(v) Compute the incremental value of the Cauchy stress and update it,

[Δσn+1]k = [B]g(−[Rn+1]k − δ(Δλn+1)k( ∂f

∂[σn+1]

)k),

[σn+1]k+1 = [σn+1]k + [Δσn+1]k,

Δλk+1n+1 = Δλk

n+1 + δ(Δλn+1)k.

Set k := k + 1, and go to Step (ii).The present example is concerned with the deformation that the changes of the three prin-

cipal values of the Cauchy-Green deformation are first calculated by the elastic constitutiveequation when the compressive stress σ1 increases and σ2 = σ3 = 0. The yield condition isthen checked by the trial elastic stress. If the value of the yield function is less than zero, thecorresponding stresses are at the elastic state; else, compute the elastoplastic tangent modulusassociated with the Drucker-Prager yield surface and update the stresses and strains by meansof the closest point projection method[10] and obtain the real stresses. The material parametersof the model are summarized in Table 1. The three trial principal values of the left Cauchy-Green deformation tensor b are calculated for 50 deformation incremental steps from the elasticconstitutive equation, as shown in Table 2.

Table 1 Material properties

E/MPa μ c/MPa φ α k/MPa

3.0 0.28 0.11 25.20◦ 0.14 0.10

Table 2 Three principal values of Cauchy-Green deformation tensor

ValueStep

1 2 3 · · · 44 45 46 47 48 49 50

b(1) 1.001 0 1.001 9 1.002 9 · · · 1.043 5 1.044 5 1.045 5 1.046 5 1.047 5 1.048 5 1.049 6

b(2) 0.999 7 0.999 5 0.999 2 · · · 0.988 2 0.987 9 0.987 6 0.987 4 0.987 1 0.986 8 0.986 6

b(3) 0.999 7 0.999 5 0.999 2 · · · 0.988 2 0.987 9 0.987 6 0.987 4 0.987 1 0.986 8 0.986 6

The typical computational results of the algorithmic tangent modulus at finite strains aregiven in Table 3, where aep(i, j) is the element of the ith row and jth column of the matrix [aep]expressed in Eq. (38). Before Step 47 of the Cauchy-Green deformation incremental, the stressstate is elastic. However, the algorithmic tangent modulus decreases slightly due to not onlythe change of the principal directions which are functions of the total deformation tensor butalso the change of the elastic modulus shown in Eq. (38) at finite strains, excluding the termsaep(2,3) and aep(3,2) which slightly increase. The stress state is plastic after Step 47. Thereare significant decreases in the diagonal terms when the stress state is changed from elasticityinto plasticity between Step 46 and Step 47, and then such diagonals slightly stably decrease.However, the off-diagonal terms slightly increase, and then slowly decrease, excluding the termsaep(2,3) and aep(3,2). In the whole stage, the algorithmic tangent modulus is not symmetric,which coincides with the expression shown in Box 1.

Similarly, the corresponding typical computational results of the elastoplastic tangent mod-ulus at finite strains are given in Table 4, where aep(i, j) is the element of the ith row andthe j column of the matrix [aep] expressed in Eq. (38). In elasticity, the elastoplastic tangentmodulus decreases slightly due to the term 1

J [σ][

∂J∂ε

]T, which provokes to the reduction of the

algorithmic tangent modulus. When the stress state is changed from elasticity to plasticity,


the diagonal terms first significantly decrease, and then decrease in a stabilized manner. Incontrast, the off-diagonal terms first slightly increase, and then slightly decrease, excludingaep(2,3) and aep(3,2). In the whole deformation process, the elastoplastic tangent modulus isasymmetrical due to the term 1

J [σ][

∂J∂ε

]T.

Table 3 Algorithmic tangent modulus at finite strains

Step aep(1,1) aep(1,2) aep(1,3) aep(2,1) aep(2,2) aep(2,3) aep(3,1) aep(3,2) aep(3,3)

1 6.580 6 2.560 0 2.560 0 2.547 4 6.595 8 2.563 5 2.547 4 2.563 5 6.595 8

2 6.561 4 2.558 1 2.558 1 2.535 0 6.589 7 2.564 5 2.535 0 2.564 5 6.589 7

3 6.542 0 2.558 1 2.558 1 2.524 4 6.585 6 2.568 0 2.524 4 2.568 0 6.585 6

· · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · ·44 5.747 4 2.535 3 2.535 3 2.017 7 6.397 5 2.675 1 2.017 7 2.675 1 6.397 5

45 5.729 0 2.535 7 2.535 7 2.006 3 6.394 3 2.678 6 2.006 3 2.678 6 6.394 3

46 5.710 7 2.536 1 2.536 1 1.994 8 6.391 0 2.682 1 1.994 8 2.682 1 6.391 0

47 5.583 4 2.967 9 2.967 9 2.753 1 3.237 4 −0.430 4 2.753 1 −0.430 4 3.237 4

48 5.582 0 2.927 7 2.927 7 2.703 1 3.164 3 −0.449 0 2.703 1 −0.449 0 3.164 3

49 5.573 9 2.902 0 2.902 0 2.669 9 3.109 5 −0.451 5 2.669 9 −0.451 5 3.109 5

50 5.558 1 2.879 1 2.879 1 2.639 0 3.064 1 −0.455 7 2.639 0 −0.455 7 3.064 1

Table 4 Elastoplastic tangent modulus at finite strains

Step aep(1,1) aep(1,2) aep(1,3) aep(2,1) aep(2,2) aep(2,3) aep(3,1) aep(3,2) aep(3,3)

1 13.171 3 5.092 9 5.096 5 5.120 0 13.198 3 5.119 8 5.120 0 5.120 0 13.198 2

2 13.142 7 5.066 4 5.073 5 5.116 3 13.192 9 5.116 5 5.116 3 5.116 3 13.192 8

3 13.114 0 5.039 4 5.050 2 5.116 3 13.191 0 5.116 3 5.116 3 5.116 3 13.191 1

· · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · ·44 11.935 3 3.956 7 4.114 2 5.070 6 13.074 8 5.070 8 5.070 6 5.070 6 13.075 0

45 11.908 1 3.932 1 4.093 0 5.071 4 13.074 2 5.071 3 5.071 4 5.071 4 13.074 2

46 11.880 9 3.907 6 4.071 8 5.072 2 13.073 6 5.071 6 5.072 2 5.072 2 13.073 3

47 11.636 1 5.431 6 5.580 9 5.935 8 6.770 4 −1.152 7 5.935 8 −1.228 9 6.846 5

48 11.641 7 5.329 5 5.483 0 5.855 5 6.679 6 −1.241 1 5.855 5 −1.318 3 6.756 8

49 11.633 8 5.261 2 5.418 4 5.803 9 6.623 0 −1.295 1 5.803 9 −1.375 7 6.701 4

50 11.613 0 5.197 2 5.358 8 5.758 2 6.573 9 −1.339 9 5.758 2 −1.420 0 6.654 0

Table 5 lists the typical computational results of the elastoplastic tangent modulus at in-finitesimal strains, where cep(i, j) is the element of the ith row and the jth column of thematrix [cep] expressed in Eq. (39). Before the Cauchy-Green deformation incremental Step 46,the stress state of material is in elasticity. The values of the elastoplastic tangent modulus areprecisely the same as those of the elastic modulus, which coincide with Eq. (39). The stressstate of material is in plasticity after Step 46. There are also significant decreases in the diag-onal terms when the stress state of material is changed from elasticity into plasticity, and thenkeeps gradually reduction excluding term cep(1,1). However, the off-diagonal terms also firstincrease, and then slightly decrease, excluding the terms cep(2,3) and cep(3,2) as well. In thewhole process, the elastoplastic tangent modulus is symmetrical.

Comparing with the computational results at finite strains in Table 4 with those at infinitesi-mal strains in Table 5, we can find that they are approximately identified. It is also interestinglyshown from the numerical results that the tendency of the general change of the elastoplastictangent modulus at finite and infinitesimal strains is similar during the deformation process.


Table 5 Elastoplastic tangent modulus at infinitesimal strains

Step cep(1,1) cep(1,2) cep(1,3) cep(2,1) cep(2,2) cep(2,3) cep(3,1) cep(3,2) cep(3,3)

1 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0

2 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0

3 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0

· · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · ·44 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0

45 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0 5.120 0 5.120 0 5.120 0 13.200 0

46 12.855 4 6.538 0 6.538 0 6.538 0 7.365 1 −0.714 9 6.538 0 −0.714 9 7.365 1

47 12.873 4 6.505 0 6.505 0 6.505 0 7.327 0 −0.753 0 6.505 0 −0.753 0 7.327 0

48 12.895 9 6.462 0 6.462 0 6.462 0 7.278 1 −0.801 9 6.462 0 −0.801 9 7.278 1

49 12.848 6 6.553 4 6.553 4 6.553 4 7.383 0 −0.697 0 6.553 4 −0.697 0 7.383 0

50 12.932 6 6.387 2 6.387 2 6.387 2 7.194 6 −0.885 4 6.387 2 −0.885 4 7.194 6

6 Conclusions

The exact tensorial forms of the algorithmic tangent modulus at finite strains are derived inthe pricipal space for three different eigenvalues. Their corresponding specific matrix expressionsare also obtained. The algorithmic tangent modulus consists of two terms, one is independentof the specific model of plasticity, and the other associates with the chosen yield surface. Thematrix expressions are detailed by establishing proper second-order tensors, as shown in Box 1.The explicit expression of the elastoplastic tangent modulus at the logarithmic strain by thelocal multiplicative decomposition is obtained by considering a general yield surface and anon-associated flow rule. The present algorithmic tangent modulus at finite strains is verifiedby a numerical example, in which the Drucker-Prager model of elastoplastic material and theassociated flow rule most commonly used in engineering analysis under the conditions of nohardening are employed.

References

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Appendix A

(i) λ1 �= λ2 �= λ3

C = λ21N

1 ⊗ N1 + λ22N

2 ⊗ N2 + λ23N

3 ⊗ N3.

Differentiating the spectral decomposition of C, we obtain

dC = 2λ1dλ1N1 ⊗ N1 + λ2

1dN1 ⊗ N1 + λ21N

1 ⊗ dN1 + 2λ2dλ2N2 ⊗ N2

+ λ22dN2 ⊗ N2 + λ2

2N2 ⊗ dN2 + 2λ3dλ3N

3 ⊗ N3 + λ23dN3 ⊗ N3 + λ2

3N3 ⊗ dN3.

Contracting the relation with N1 ⊗ N1 and using NA · NA = 1 and dNA · NA = 0 for A = 1, 2, 3,we obtain

∂λ1

∂C=

1

2λ1N1 ⊗ N1.

Similarly,

∂λ2

∂C=

1

2λ2N2 ⊗ N2,

∂λ3

∂C=

1

2λ3N3 ⊗ N3.

(ii) λ = λ1 = λ2 �= λ3

Similar to (i), we obtain

∂λ3

∂C=

1

2λ3N3 ⊗ N3, 2

∂λ

∂C=

1

2λ(N1 ⊗ N1 + N2 ⊗ N2).

Since

C−1 = λ−2N1 ⊗ N1 + λ−2N2 ⊗ N2 + λ−23 N3 ⊗ N3,

we obtain

2∂λ

∂C=

1

2λ(C−1 − M3).

(iii) λ = λ1 = λ2 = λ3

Similar to (ii), we obtain

3∂λ

∂C=

1

2λC−1.

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Algorithmic tangent modulus at finite strains based on multiplicative decomposition