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Tangent modulus tensor in plasticity under finite strain

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Summary

The tangent modulus tensor, denoted as\(\mathfrak{D}\), plays a central role in finite element simulation of nonlinear applications such as metalforming. Using Kronecker product notation, compact expressions for\(\mathfrak{D}\) have been derived in Refs. [1]–[3] for hyperelastic materials with reference to the Lagrangian configuration. In the current investigation, the corresponding expression is derived for materials experiencing finite strain due to plastic flow, starting from yield and flow relations referred to the current configuration. Issues posed by the decomposition into elastic and plastic strains and by the objective stress flux are addressed. Associated and non-associated models are accommodated, as is “plastic incompressibility”. A constitutive inequality with uniqueness implications is formulated which extends the condition for “stability in the small” to finite strain. Modifications of\(\mathfrak{D}\) are presented which accommodate kinematic hardening. As an illustration,\(\mathfrak{D}\) is presented for finite torsion of a shaft, comprised of a steel described by a von Mises yield function with isotropic hardening.

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Abbreviations

B:

strain displacement matrix

C=FTF:

Green strain tensor

\(\mathfrak{C},\mathfrak{C}_e ,\mathfrak{C}_p\) :

compliance matrix

D=(L+LT)/2:

deformation rate tensor

D :

fourth order tangent modulus tensor

\(\mathfrak{D},\mathfrak{D}_e ,\mathfrak{D}_p\) :

tangent modulus tensor (second order)

d :

VEC(D)

e :

VEC(ɛ)

E :

“Eulerian pseudostrain”

F, F e ,F p :

Helmholtz free energy

F=∂x/∂X:

deformation gradient tensor

f :

consistent force vector

ℊ:

residual function

G:

strain displacement matrix

h :

history vector

h :

time interval

H :

function arising in tangent modulus tensor

I, I 9 :

identity tensor

i :

VEC(I)

k 0,k 1 :

parameters of yield function

K g :

geometric stiffness matrix

K T :

tangent stiffness matrix

k k :

kinematic hardening coefficient

J :

Jacobian matrix

L=∂v/ϖx :

velocity gradient tensor

m :

unit normal vector to yield surface

M:

strain-displacement matrix

N:

shape function matrix

n :

unit normal vector to deformed surface

n 0 :

unit normal vector to undeformed surface

n :

unit normal vector to potential surface

r, R, R 0 :

radial coordinate

s :

VEC(δ)

S :

deformed surface

S 0 :

undeformed surface

t :

time

t, t 0 :

traction

t :

VEC(τ)

\(\mathop t\limits^ \circ\) :

VEC(τ\(\mathop \tau \limits^ \circ\))

t′:

VEC(τ′)

t r :

reference stress interior to the yield surface

t :

t−t r

T :

kinematic hardening modulus matrix

u=x−X :

displacement vector

U:

permutation matrix

v=∂x/∂t :

particle velocity

V :

deformed volume

V 0 :

undeformed volume

X :

position vector of a given particle in the undeformed configuration

x(X,t):

position vector in the deformed configuration

z, Z :

axial coordinate

γ:

vector of nodal displacements

ɛ=(FTF−I)/2:

Lagrangian strain tensor

η:

history parameter scalar

θ, Θ:

azimuthal coordinate

ϰ:

elastic bulk modulus

λ:

flow rule coefficient

Λ:

twisting rate coefficient

μ:

elastic shear modulus

ν:

iterate

σ:

Second Piola-Kirchhoff stress

τ:

Cauchy stress

\(\mathop \tau \limits^ \circ\) :

Truesdell stress flux

τ′:

deviatoric Cauchy stress

Y, Y′ :

yield function

ϕ:

residual function

Φ:

plastic potential

X, Xe, Xp :

second order tangent modulus tensors in current configuration

X′, Xe′, Xp′ :

second order tangent modulus tensors in undeformed configuration

δ(.):

variational operator

VEC(.):

vectorization operator

TEN(.):

Kronecker operator

tr(.):

trace

⊗:

Kronecker product

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Author notes

  1. D. W. Nicholson & B. Lin Ph.D.

    Present address: Institute for Computational Engineering, Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, 32816, Orlando, FL, USA

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    1. D. W. Nicholson
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    2. B. Lin Ph.D.
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    Nicholson, D.W., Lin, B. Tangent modulus tensor in plasticity under finite strain. Acta Mechanica 134, 199–215 (1999). https://doi.org/10.1007/BF01312655

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