Constitutive Assumptions and Equations

Modeling of the Thermomechanical Behavior, Damage, and Fracture of High

22.2 Thermomechanical Framework

22.2.2 Constitutive Assumptions and Equations

The high formability and ductile damage mechanisms of TRIP-steels imply a finite deformation framework as basis of constitutive modeling. As constitutive assump- tion, an additive split of the rate of deformation tensorDinto an elastic, viscoplastic, and transformation induced part is considered:

D= D_el+D_vpl+D_trip. (22.17) A state potential based on the logarithmic strain is proposed:

ψ=ψ

ε^el_log, ϑ, κ_α, εnl,grad_xεnl

. (22.18)

The constitutive-dependent quantities

σ,s,K_α,m_nl,M_nl

have the same depen- dencies asψ(principle of equi-presence). Utilizing theColeman-Noll-procedure [25], constitutive relations can be deduced from the dissipation inequality (22.16) plugging in the state potential (22.18):

σ =ρ ∂ψ

∂ε^el_log,s= −∂ψ

∂ϑ, M_nl=ρ ∂ψ

∂grad_xεnl,m_nl=ρ∂ψ

∂εnl. (22.19) The conjugated thermodynamic forces to the internal variables are defined as

K_α=ρ∂ψ

∂κ_α. (22.20)

Furthermore, expressions for the thermal and mechanical dissipation remain, which have to be fulfilled independently:

γth= −1

ϑq_th·grad_xϑ≥0, (22.21)

γm=σ :

Dvpl+Dtrip

−K_ακ˙α ≥0. (22.22)

The thermal and mechanical dissipation inequalities set restrictions on the definitions of the heat flux vectorq_th, the evolution equations of the inelastic deformation rates

D_vpl,D_trip

, and the internal variablesκαto be proposed.

22.2.2.1 State Potential

To specify the state potential (22.18), the set of state variables is complimented by the choice of scalar internal variablesκ_α= {r, fsb,z,D}:

• r: strain hardening variable (isotropic)

• fsb: volume fraction of shear bands

• z: volume fraction ofα-martensite

• D: damage variable (isotropic)

Subsequently, a multiplicative viscoplasticity approach is introduced, as classified by Lemaître and Chaboche [26, Sect. 6.4.2]. Therefore, the state potential must not contain the hardening variablerto ensure non negative mechanical dissipation, see [26, Sect. 6.4.2]. Moreover, the stored energy in shear bands is neglected. Thus, the Helmholtzfree energy is given asψ

ε^el_log, ϑ,z,D, εnl,grad_xεnl

: ρψ=ρψel

ε^el_log, ϑ,D

+ρψchem(ϑ,z)+ρψϑ(ϑ)+ρψnl

εnl,grad_xεnl

. (22.23) The thermoelastic part reads

ψel=(1−D)1 ρ 1 2

ε^el_log−αth(ϑ−ϑ0)δ :C:

ε^el_log−αth(ϑ−ϑ0)δ . (22.24) Isotropic thermoelastic properties are assumed for the cast TRIP-steel, defined by the fourth order tensor of elastic stiffnessC and the thermal expansion coefficient αth. Moreover, the elastic and thermal expansion properties are assumed to be similar for the austenitic and martensitic phase [27]. DamageDdegrades the elastic energy leading to material softening.

The chemical part describes the difference inGibbs-energy of the two phases given by a rule of mixture:

ψchem=(1−z) ψa(ϑ)+zψm(ϑ) . (22.25)

The purely temperature dependent term ψ_ϑ =c⁰

ϑ−ϑln ϑ

ϑ∗

, (22.26)

defines a constant partc⁰of the specific heat similar for both phases with a reference temperatureϑ_∗.

The last term in (22.23) comprises the micromorphic contributions. A conven- tional quadratic structure is prescribed, see [23]:

ρψnl=1 2H_nl

⎛

⎝ t t₀

˙ εl

D_vpl

dτ −εnl

⎞

⎠

2

+1

2Anlgrad_xεnl·grad_xεnl. (22.27) The first part penalizes the difference of the local strain like value, defined by the integral, and the micromorphic variableεnl. The penalty stiffness is prescribed by the parameter Hnl. The integral defines a scalar value, which is extracted from the viscoplastic deformation rate D_vplserving as driving force for damage. The second term states the micromorphic gradient contribution weighted by the parameterA_nl.

22.2.2.2 Consistent Rate Formulation

TheCauchy-stress defined in (22.19) can be evaluated using (22.23) and (22.24):

σ =ρ ∂ψ

∂ε^el_log =(1−D)C:

ε^el_log−αth(ϑ−ϑ0)δ

. (22.28)

Applying the effective stress concept [28], the constitutive equation reads ˆ

σ = σ

1−D =C:

ε^el_log−αth(ϑ−ϑ0)δ

, (22.29)

whereσˆ designates the effective stress acting on an undamaged cross section of the material;σ is then interpreted as the net stress. In what follows, a hat(·)ˆ highlights quantities which are computed from the effective stress tensorσˆ.

The stress-strain relation has to be given in an objective rate form in order to imple- ment the model into the user subroutine UMAT of the FEM-software ABAQUS.

As objective time derivative, the logarithmic rate (·) introduced by Xiao et al.

[29,30] is applied to both sides of (22.29). Hence, the rate formulation

ˆ σ =C:

D_el−αinstδϑ˙

, (22.30)

is consistent with the notion of elasticity [31], whereαinstdenotes the instantaneous coefficient of thermal expansion. The logarithmic rate of the logarithmic strain tensor ε^el_logyields the rate of deformation tensor Delas shown in [30]. The logarithmic rate of the effectiveCauchy-stress reads

ˆ σ = dσˆ

dt −log·σˆ −σˆ ·^T_log. (22.31) The skew logarithmic spinlogcan be found in [29].

22.2.2.3 Generalized Stresses

TheColeman-relations for generalized stresses in (22.19) are specified as

M_nl=A_nlgrad_xεnl, (22.32)

mnl=Hnl

⎛

⎝εnl− t t0

˙ εl

Dvpl

dτ

⎞

⎠=Hnl(εnl−εl) , (22.33)

where the integral is substituted for reasons of clarity byεl. With (22.32) and (22.33) at hand, the micromorphic balance (22.12) can be rewritten in the established manner [23,32]

L²_nlxεnl=εnl−εl ∀x∈B, (22.34) where the internal lengthLnlis defined as

L²_nl= A_nl

H_nl ≥0. (22.35)

The spatial Laplacian is denoted asx. Thereby, the micromorphic balance (22.34) is identical to the partial differential equation of implicit gradient enhancement pro- posed by Peerlings et al. [33], which has been used in own preliminary studies [34, 35]. Following the suggestion in [33], trivial natural boundary conditions are pre- scribed on all free boundaries

grad_xεnl· n=0 ∀x∈∂Bm=∂B. (22.36) As result of this choice, the overall mean values of the micromorphic variable and its local counterpart are equal [34].

22.2.2.4 Thermodynamic Forces

The conjugated thermodynamic forcesK_α= {R,Fsb,Z,Y}associated with the set of internal variables are specified by (22.20) as

R=ρ∂ψ

∂r =0, (22.37)

Fsb=ρ∂ψ

∂f_sb =0, (22.38)

Z =ρ∂ψ

∂z =ρψm(ϑ)−ρψa(ϑ)= g_chem^a→m(ϑ) , (22.39) Y =ρ∂ψ

∂D = −1 2

ε^ellog−αth(ϑ−ϑ0)δ :C:

ε^ellog−αth(ϑ−ϑ0)δ . (22.40) The driving force for martensitic phase transitionZconsists of the energy difference g^a_chem^→m(ϑ), which is negative once the temperature is below the thermodynamic equilibrium temperature [36], i.e., in the region of interest. The energy release rate Y is always negative or zero,Y ≤0.