al elasto-p astic and reep material models

•

independent, and the ma ould alternatively have c

m

al elasto-p astic and reep material models

• This section groups together thermal elasto-plastic materials and reep materials, since a unified general solution can be applied to c

Isotropic elastic strains, via the MAT1 entry Thermal strains, ^{t TH}e_rs , via the MATT1 or the MAT

t p rs

Time-dependent creep strains, ^{t C}e_rs, via the CREEP entry, or the CREEP and MATTC entries.

• tensor at the temperature corresponding to tim . The tensor can be expressed in terms of Young's modulus ^tE and Poisson's

• Note that the thermal, plastic and creep parts of these material models are optional. If, however, the omitted strain components result in one of the material models detailed in one of the previous

ref. KJB Section 6.6.

t E

Cijrs

e t ^tC_ijrs^E ratio ^tv both of which may be temperature-dependent.

3

sections, then the program will select that material model.

• The formulations provided in this section are very general, and an describe any material combining elastic, plastic, thermal and

ns given in Table 3.6-1 are allowed.

, 3-D

ent/small strain, large displacement/small strain and large displacement/

large strain kinematics.

When used with small displacement/small strain kinematics, a materially-nonlinear-only formulation is employed.

When used with large displacement/small strain kinematics, either a TL or a UL formulation is employed (TL for 2-D and 3-D

When used with large displacement/large strain kinematics, the s c

creep strains. The combinatio

• These material models can be used with the rod, 2-D solid solid, and shell elements.

• These models can be used with small displacem

solids and shells, and UL for rods).

ULH (updated Lagrangian Hencky) formulation is employed. Thi is only supported for 2-D solid and 3-D solid elements.

• Plane strain, axisymmetric or 3-D solid elements that reference these material models should preferably employ the mixed u/p element formulation. This is done by setting UPFORM =1 in the NXSTRAT entry.

Description Elastic Plastic Creep Bulk data entries

Yes No Temp-dep

MAT1, CREEP, MATTC

Yes Yes No MAT1, M TS1, wA ith TID in MATS1 pointing to a TABLES1 entry

Tem

dep with TID in MATS1

pointing to a TABLES1 entry p- Yes No MAT1, MATT1, MATS1, Yes

Temp-dep

No MAT1, MATS1, wi h TID in MATS1 pointing to a TABELST entry

t dep pointing to a TABELST entry

Yes Yes Yes MAT1, MATS1, CREEP, with TID in MATS1 pointing to a TABLES1 entry

Plastic-creep

Tem

dep p- Yes Yes MAT1, MATT1, MATS1, CREEP, with TID in MATS1 pointing to a TABLES1 entry

Yes Temp-ep

Yes MAT1, MATS1, CREEP, with TID in MATS1 pointing to a TAB ry

d LEST ent

Tem

dep

p-dep Yes MAT1, MATT1, MATS1, CREEP, with TID in MATS1 pointing to a TABLEST entry

p- Tem

Yes Yes

Temp-dep MAT1, MATS1, CREEP, MATTC, with TID in MATS1 pointing to a TABLES1 entry

Tem

dep p- Yes

Temp-dep MAT1, MATT1, MATS1, CREEP, MATTC, with TID in MATS1 pointing to a TABLES1 entry

Yes Tem

p p- Temp- MAT1, MATS1, CREEP, MATTC, with TID in de dep MATS1 pointing to a TABLEST entry

Thermal

Temp-dep MAT1, MATT1, MATS1, CREEP, MATTC, with TID in MATS1 pointing to a TABLEST entry

Notes:

. "No" means that this strain is not included in the material. "Yes" means that this strain is included in the mat

temperature-indepe description, and tha

Table 1

erial description, and that the material constants for this strain are ndent. "Temp-dep" means that this strain is included in the material t the material constants for this strain are temperature-dependent.

3.6-1: Combinations of elastic, plastic and creep strains

creep strains are independent of each other; hence the only interaction between the strains comes from the fact that all strains affect the stresses according to Eq. 3

constitutive description for a one-dimensional stress situation and a bilinear stress-strain curve.

• Note that the constitutive relations for the thermal, plastic and

.6-1. Fig. 3.6-1 summarizes the

Area A

l l

tP Temperature^tq

(a) Model problem of rod element under constant load

Stress

ts

tE ( )Ttq

tE( )tq

tsyv^{( )t}q

t Pe (time independent)

t Ee (time independent) Strain

Creep strain te (time dependent)C

t Time

(b) Strains considered in the model

Figure 3.6-1: Thermo-elasto-plasticity and creep constitutive description in one-dimensional analysis

• multilinear plasticity, the rupture plastic strain corresponds to strain at the last point input for the stress-strain r

oint, ding element is removed from the model (see Section 0.5).

ref. M.D. Snyder and K.J. Bathe, "A Solution Procedure for Thermo-Elastic-Plastic and Creep Problems," J. Nuclear Eng. and Design, Vol. 64, pp. 49-80, 1981.

ref. M. Kojić and K.J. Bathe, "The Effective-Stress-Function p," Int.

h. ngn -1532,

3.6.1 Eval

rmal strains are calculated as described in Section 3.1.6.

3.6.2 Eval

• Plasticity effects are included in the thermal elasto-plastic on the vo s yield criterion, an

ening (no mixed hardening), and bilinear or multilinear stress-strain curves (based on the H and TID fields in MATS1).

• In the case of bilinear plasticity only the elastic material

s case, the ting Since there is no direct coupling in the evaluation of the different strain components, we can discuss the calculation of each strain component independently.

In

the effective plastic cu ve.

• When rupture is reached at a given element integration p the correspon

1

Algorithm for Thermo-Elasto-Plasticity and Cree J. Numer. Met E g., Vol. 24, No. 8, pp. 1509 1987.

uation of thermal strains

• The the

uation of plastic strains

material model and is based n Mise associated flow rule, isotropic or kinematic hard

parameters can be temperature dependent (Young’s modulus, Poisson’s ratio and coefficient of thermal expansion).

• The multilinear plasticity case is more general. In thi stress-strain curves can be made temperature dependent by set TID in MATS1 to point to a TABLEST entry instead of a TABLES1 entry. The elastic material parameters can also be

tem erature dependent. The yield curves are interpolated as shown p in Fig. 3.6-2.

a) Stress-strain curves input data

b) Yield curves Stress

1

4

1

2

2

3

3 4

e¹_i e_i² e_i³ e_i⁴ Strain

i+1

1 3 4

qi

q

e_i+1 e_i+1² e_i+1 e_i+1 Temperature

Stress

e e

Temperature ^{i+1 i+1}

P2 P3

Interpolated yield curve

Plastic strain 2

tq qi

3 4

1

1 2 3 4

qi+1

Figure 3.6-2: Interpolation of multilinear yield curves with temperature

The plastic strains are calculated using the von Mises plasticity (Young's modulus, Poisson's ratio, stress-strain curves).

• yie

•

model (see Section 3.4) with temperature-dependent material parameters

The ld function is in isotropic hardening

1 1 2

2 3

t t t t

fy = s s⋅ −

σ

_yv

and in kinematic hardening

( ) ( )

²v

o ue to kinematic hardening.

tic str rements resulting from the stress corresponding to temperature ^tθ d ^tα is the shift of stress tens r d

• The expressions for plas ain inc

flow theory are de_ij^P =d

λ

^ts_ij for isotropic hardening and

(

^t ij

)

P t

ij ij

de =d

λ

s −

α

for kinematic hardening, in which dλ is the plastic multiplier (positive scalar) which can be determined from the yield condition ^tf_y = 0. In the case of kinematic hardening, we express the change of the yield surface position in the form

P

ET as a function of temperature is large. Hence, it is recommended to use this model only when the variation in ET as a

s small.

3.6.3 Eval t

currently available in Solution 601. The first called the Power creep law is obtained by setting TYPE = 300 in the CREEP material entry. The second creep law called the Exponential creep law is obtained by setting TYPE = 222 in the CREEP material entry. The Power creep law is currently supported for the elastic-creep, thermal elastic-creep, plastic-creep and

e Exponential creep la he elastic-creep and plastic-creep

• Care should be exercised in the use of this model in kinematic hardening conditions. Namely, nonphysical effects can result when the variation in

function of temperature i ua ion of creep strains

• Two creep laws are

thermal plastic-creep material models. Th w is currently supported only for t

aterial models.

m

• The effective creep strain is calculated as follows:

Power creep law (creep law 1) :

teC = ⋅a t

σ

b⋅td

in which σ is the effective stress, t is the time, and a, b, d are material constants from the CREEP material entry. These three

Exponential creep law (creep law 2) :

constants can be set to be temperature dependent via the MATTC entry.

( ) (

^{1 R( )t}

) ^{( )}

teC =F σ ⋅ −e^{− σ ⋅} + σ ⋅G t with

( ) ( )

^{bt ;}

^{( ) ( )}

^{t d;}

^{( )} ( )

^t

F σ = ⋅a e ^{⋅ σ} R σ = ⋅ σc G σ = ⋅e e^f⋅σ

in which a through f are material constants from the CREEP material entry.

The creep strains are evaluated using the strain hardening procedure for load and temperature variations, and the O.R.N.L.

r cyclic loading conditions.

ref. C.E. Pugh, J.M. Corum, K.C. Liu and W.L. Greenstreet,

"Currently Recommended Constitutive Equations for Inelastic Design of FFTF Components," Report No. TM-3602, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 1972.

and

• rules fo

The procedure used to evaluate the incremental creep strains is summarized in the following: Given the total creep strains ^{t C}e_ij the deviatoric stresses ^t+∆ts_ij,

1) Calculate the effective stress

1

3 2

2

t t t t t t

ij ij

s s

σ

+∆ ⎡ +∆ +∆ ⎤

= ⎢⎣ ⎥⎦

2) Calculate the pseudo-effective creep strain

( )( )

¹²

2 3

t C t C orig t C orig

ij ij ij ij

e =⎡⎢⎣ e −e e −e ⎤⎥⎦

3a) For power creep with temperature-independent material constants, calculate the effective creep strain and effective creep strain rate at time t+ ∆tusing

( )

²

( )

²

(

0 ¹

)

²

a

t^+∆teC ^1/a = teC ^1/a + a t^+∆tσ ^1/a ∆t

t t t C

t t C e e

e t

+∆ +∆ −

= ∆

 ^C

3b) For other creep laws (including power creep with

temperature dependent constants), calculate the pseudo-time t satisfying

( ) ( )

C t t t t t C C t t t t

e ^+∆

σ

,^+∆

θ

,t = e +e ^+∆

σ

,^+∆

θ

,t ∆t

where ^eC

(

^t+∆t

^σ

^,t+∆t

^θ

^,t

)

is the generalized uniaxial creep law and _C

(

_{t t t t}

)

^{d eC}

(

^t+∆t

^σ

^t+∆t

^θ

^t

)

e

σ θ

t

d t

+∆ +∆ , ,

, , =

 

. Then calculate the effective creep strain and effective creep strain rate at time

t+ ∆t using

( )

t+∆teC =eC t+∆t

σ

,t^+∆t

θ

,t , ^{t+∆teC =eC}

(

^t+∆t

^σ

^,t+∆t

^θ

^,t

)

^.

4) Calculate ^t+∆t

γ

using

t^+∆teC

3 

t^+∆t

γ

= _{t t}

σ

2 +∆

5) Calculate the incremental creep strains using

C t t t t

ij ij

e t γ s

∆ = ∆

The use of the pseudo-time in step 3b corresponds to a strain hardening procedure. See ref. KJB, pp 607-608 for a discussion of strain hardening for calculation of creep

+∆ +∆

strains.

3.6.4 Com

ints are evaluated

-1532,

riefly, the procedure used consists of the following calculations.

ref. KJB Section 6.6.

putational procedures

• The stresses and strains at the integration po using the effective-stress-function algorithm.

ref. M. Kojić and K.J. Bathe, "The Effective-Stress-Function Algorithm for Thermo-Elasto-Plasticity and Creep," Int.

J. Numer. Meth. Engng., Vol. 24, No. 8, pp. 1509 1987.

B

The general constitutive equation

3

( )

( ) ( ) ( ) ( )

t^+∆t

σ

i =t^+∆tCE t^+∆tei −t^+∆t P ie −t^+∆t C ie −t^+∆t THe (3.6-1) ean stress and for the deviatoric

dex will be dropped in the discussion to follow. The mean stress calculated as

is solved separately for the m

stresses. In this equation the index (i) denotes the iteration counter n the iteration for nodal point equilibrium. For easier writing this i they can be expressed as

t t 1 t t e step and α is the integration parameter used for stress tim

evaluation

(

^{0≤ <α 1}

)

. The creep and plastic multipliers ^τ

γ

and λ

∆ are functions of the effective stress ^t+∆t

σ

only, and they ccount for creep and plasticity

a ; also

t+∆te′′= t+∆te′−t Pe −t Ce

is known since the deviatoric strains ^t+∆te′, plastic strains ^{t P}e and reep strains ^{t C}e are known from the current displacements and c

the stress/strain state at the start of the current time step.

lowing scalar function

The fol ^f

(

^t+∆t

^σ )

is obtained from Eq.

(3.6-3)

(

^{t t}

)

^{2t t 2 0}

f ^+∆

σ

=a ^+∆

σ

²+b^τ

γ

−c ^τ

γ

²−d² = (3.6-4)

The zero of Eq. (3.6-4) provides the solution for the effective stress

with s ation On e sol

umm on the indices i, j.

ce th ution for has been determined from Eq. (3.6-4), ultaneously with the scalars ^τ

γ

and ∆λ

sim from the creep and

lasticity conditions, the deviatoric stress is calculated from q. (3.6-3), and the plastic and creep strains at the end of the time

re obtained as

d shell elements) are given in the above cited ferences, and also in the following reference:

ctures, Vol. 26, No 1/2, pp. 135-143, 1987.

The above equations correspond to isotropic hardening conditions and a general 3-D analysis. The solution details for kinematic hardening conditions and for special problems (for the plane stress an

re

ref. M. Kojić and K.J. Bathe, "Thermo-Elastic-Plastic and Creep Analysis of Shell Structures", Computers &

Stru

3.7 Hyperelastic material models

The hyperelastic material models available in Advanced onlinear Solution are the Mooney-Rivlin, Ogden, Arruda-Boyce,

-Bath material models. They are all E com and. In addition MATHP can be

a

e 2-D solid and solid elements.

el uses

ed. The e

• Viscoelastic effects and Mullins effects can be included using e MATHEV and MATHEM entries.

xpansion coefficient. Section 3.7.6 shows how they are computed

• In Solution 701 only the Mooney-Rivlin and Ogden material

• The isotropic hyperelastic effects are mathematically described y specifying the dependence of the strain energy density (per unit rigi l volume) on the Green-Lagrange strain tensor

• N

Hyperfoam, and Sussman e efined using the MATH m d

used to define a hyperelastic Mooney-Rivlin m terial.

• This material model can be employed with th D

• This material mod large displacement/large strain kinematics. A Total Lagrangian (TL) formulation is employ same formulation is used if a large displacem nt/small strain kinematics is selected.

th

• Thermal strains can be included via a constant thermal e

for hyperelastic materials.

models can be used, and only for 3-D solid elements.

b

W

ε

_ij

o na .

We now give a brief summary of the quantities and concepts .6.2

, iven by

•

used. For more information, refer to ref KJB, section 6 . Here and below, we omit the usual left superscripts and subscripts for ease of writing. Unless otherwise stated, all quantities are evaluated at time t and referred to reference time 0.

• Useful quantities are the Cauchy-Green deformation tensor C_ij g

ij 2 ij ij

C =

ε

+

δ

where

δ

_ij is the Kronecker delta; the principal invaria f the auchy-Green deformation tensor,

nts o C

1 kk

I =C , 2

(

1²

)

1

2 ^{ij ij}

I = I −C C , I₃ =detC

uced invariants:

the red

1 1 1 3 3

I =I I ⁻ , I₂ =I I_{2 3}⁻²³, J =I₃¹², e stretches

λ

_i

th where the

λ

_i’s are the square roots of the principal stretches of the Cauchy-Green deformation tensor; and the reduced stretches:

λ

i =

λ λ λ λ

i

(

1 2 3

)

⁻¹³

In document NX Nastran 8 Advanced Nonlinear Theory and Modeling Guide (Page 134-147)