rousselier_poly

Description

This behavior is a model developed within the self-consistent polycrystalline framework intended for taking into account various mechanisms. The framework of physically based polycrystalline metal plasticity has intrinsic advantages in describing the anisotropy and distortion of the yield surface, as well as realistic anisotropic hardening. It enables the prediction of complex behaviors in multi-axial and multi-path loadings.

The mathematical framework of polycrystalline plasticity is developed within a reduced texture methodology (RTM) in order to dramatically increase the computational efficiency while keeping the physically-based constitutive equations (Rousselier et al., 2009, 2010, 2012; Luo and Rousselier, 2014). The RTM can make use of any polycrystalline plasticity model with a very small number of crystallographic orientations. These orientations are determined from mechanical experiments through inverse optimization, in place of crystallographic measurements (EBSD).

In order to model ductile fracture, macroscopic void nucleation and growth models have been reformulated in the polycrystalline framework. Various models are introduced at the slip system scale: nucleation and growth models of a second population of very small voids, the Coulomb fracture model, dynamic strain aging model (DSA) and texture evolution.

Syntax

The basic input syntax here is:

***behavior rousselier_poly [ modifier ] \(~\,\) **elasticity <ELASTICITY> \(~\,~\,~\,\) … \(~\,\) **elasticity0 <ELASTICITY> #broken behaviour young, poisson \(~\,~\,~\,\) … \(~\,\) **grain_number … #define grain number \(~\,\) **system_type fcc | bcc #choose crystal structure \(~\,\) **orientation_file filename #load grain orientation file \(~\,\) **system_file filename #load slip system file \(~\,\) **back_stress_file_[...] filename #load back stresses file [ **hardening #define hardening law \(~\,~\,\) *isotropic <ISOTROPIC_HARDENING> \(~\,~\,\) *matrix_h_[ fcc | bcc ] \(~\,~\,~\,\) … \(~\,\) **hardening \(~\,~\,\) *isotropic <ISOTROPIC_HARDENING> \(~\,~\,\) *matrix_h_[fcc | bcc] \(~\,~\,~\,\) … ] [ **beta_rule scalar | cs_dt | ct_dt #choose beta rule type \(~\,~\,~\,\) … ] \(~\,\) **use_MC #Mohr-coulomb \(~\,\) **use_Rdamage #Rousselier damage model \(~\,~\,~\,\) … \(~\,\) **model_coef \(~\,~\,~\,\) coefs

A phenomenological Norton-like viscoplastic model (Cailletaud, 1992) is used for the constitutive equations of each slip system. The hardening term \(r_{s}\) in each slip system is described by:

(298)\[r_{s} = R_{0}+Q_{1}\sum \limits_{{t=1}}^M H_1^{st}[1-\exp(-b_{1}v_{t})]+Q_{2}\sum \limits_{{t=1}}^M H_2^{st}[1-\exp(-b_{2}v_{t})]\]

where \(H_{1}\) and \(H_{2}\) are constant hardening matrices modeling the interactions between the slip systems. With two matrices, the self and latent hardening rates of the slip systems evolve with deformation. For BCC materials, we need to provide \(h_0\) and \(h_1\). For FCC materials, \(h_0\), \(h_1\), \(h_2\), \(h_3\), \(h_4\), \(h_5\) need to be specified.

Example

For BCC materials:

**hardening \(~\,\) *isotropic nonlinear \(~\,~\,\) Q  69.9 \(~\,~\,\) b  23.2 \(~\,\) *matrix_h_bcc \(~\,~\,\) h0 1.0 \(~\,~\,\) h1 0.1 **hardening \(~\,\) *isotropic nonlinear \(~\,~\,\) Q  49.7 \(~\,~\,\) b  2.6 \(~\,\) *matrix_h_bcc \(~\,~\,\) h0 1.0 \(~\,~\,\) h1 0.1

In each ‘grain’ \(g\) of volume fraction \(f_{g}\), the stress \(\ten \sigma_{g}\) and strain \(\bf eto=\bf eel+\bf evp\) are assumed to be homogeneous. The macroscopic stress and strain are defined as spatial averages of the N grains. To establish a unique relationship between the stresses at the macroscopic and microscopic scale, the so-called \(\beta\)-rule (Cailletaud, 1992; Sai et al., 2006) is adopted here.

(299)\[\ten \sigma_{g} = \bf \Sigma + C (\ten \beta-\ten \beta_{g}) \quad \text{with} \quad \ten \beta=\sum \limits_{{g=1}}^N f_{g}\ten \beta_{g}\]

(300)\[\dot{\ten \beta}_{g} = \dot{\bf \varepsilon}_{vpg}-\bf D:\ten \beta_{g}||\dot{\bf \varepsilon}_{vpg}|| \quad \text{where} \quad ||\dot{\bf \varepsilon}_{vpg}||=\sqrt{\frac{2}{3}\dot{\bf \varepsilon}_{vpg}:\dot{\bf \varepsilon}_{vpg}}\]

where C is a scalar. The fourth order tensor \(\bf D\) is proposed by Sai et al. (2006) for anisotropic materials. With the Voigt notation, \(\bf D\) has 10 independent elements for orthotropic materials. The current module offers 3 types of \(\beta\)-rule by choosing different key words: scalar, cs_dt and ct_dt. ‘scalar’ is used when \(C\) and \(D\) are both scalars. ‘cs_dt’ is used when \(C\) is a scalar and \(\bf D\) is a tensor. ‘ct_dt’ is used when \(\bf C\) and \(\bf D\) are both tensors.

Example

For cs_dt \(\beta\)-rule:

**beta_rule cs_dt \(~\,\) *C 5.4e+04 \(~\,\) *Dtensor \(~\,~\,\) gr11 100.0 \(~\,~\,\) gr21 0.0 \(~\,~\,\) gr22 100.0 \(~\,~\,\) gr23 0.0 \(~\,~\,\) gr31 0.0 \(~\,~\,\) gr32 0.0 \(~\,~\,\) gr33 100.0 \(~\,~\,\) gr44 100.0 \(~\,~\,\) gr55 100.0 \(~\,~\,\) gr66 100.0

Other models such as macroscopic Rousselier damage model, secondary void nucleation, Coulomb fracture model and dynamic strain aging model (DSA) etc. can be activated using the key words **use_[...] defined before **model_coef. The associated parameters can then be defined after **model_coef.

Macroscopic porous plasticity

Porous plasticity is coupled with polycrystalline plasticity. The aggregate contains voids with variable volume fraction \(f\). The volume fractions of the matrix grains are \((1-f)f_{g}\). The following plastic potential is used:

(301)\[F(\frac{\ten \sigma}{1-f}) = \frac{\sigma_{eq}}{1-f}-\left(\sum \limits_{{g=1}}^N f_{g}\ten \sigma_{g}\right)+D_{1}f\sigma_{1}\exp\left(\frac{\sigma_{m}^{*}}{(1-f)\sigma1}\right)\]

where \(\sigma_{m}^{*}=\alpha_{L}\ten \sigma_{LL}+\alpha_{T}\ten \sigma_{TT}+\alpha_{N}\ten \sigma_{NN}\) with \(\alpha_{L}+\alpha_{T}+\alpha_{N}=1\). For isotropic void growth, \(\alpha_{L}=\alpha_{T}=\alpha_{N}=1/3\). In the current version, the void volume fraction rate \(\dot{f}\) is the sum of a first term due to the mass conservation law and a second term for void nucleation. The nucleation term involves 3 parameters \(f_{N}\),\(\sigma_{N}\),\(\epsilon_{N}\) (Chu and Needleman, 1980). It is worth noting the material is considered as ‘broken’ when the void volume fraction exceeds an ultimate value \(f_{u}\).

Example

**use_Rdamage **model_coef \(~\,~\,\) f0,D1,s1,aL,aT,aN,fN,eN,sN,fu

Porous plasticity at the slip system

A second population of very small voids that nucleate and grow in shear band can be considered in the current model. These voids can contribute to void coalescence of the larger voids with the so-called ‘void sheet mechanism’. An additional void volume fraction variable \(f_{s}\) is considered for each slip system. The contribution of all grains is added to \(F\).

(302)\[F(\frac{\ten \sigma}{1-f}) = \frac{\sigma_{eq}}{1-f}-\left(\sum \limits_{{g=1}}^N f_{g}\ten \sigma_{g}\right)+D_{1}\left(f+\sum \limits_{{g=1}}^N f_{g}\left(\sum \limits_{{s=1}}^M f_{s}\right)_{g}\right)\sigma_{1}\exp\left(\frac{\sigma_{m}^{*}}{(1-f)\sigma1}\right)\]

The void fraction rate could take a similar form to the macroscopic one.

Example

**use_void_meso **model_coef \(~\,~\,\) f02,D12,t12,fN2,eN2,sN2

Coulomb fracture model

The Coulomb fracture model can be used to simulate the transgranular shear fracture without voids. This mechanism is observed in some metals. To integrate this phenomenon, an additional slip rate \(\dot{\gamma}_{s}^{C}\) is introduced, so that the total slip rate is \(\dot{\gamma}^{tot} = \dot{\gamma}_{s}+\dot{\gamma}_{s}^{C}\).

Example

**use_MC **model_coef \(~\,~\,\) c3,r3,Q3,b3,p3,gvC

Dynamic strain ageing model

Dynamic strain ageing is the pinning of dislocations by solute atoms that diffuse during straining. The so-called KEMC model is added to the hardening equation of each slip system.

Example

**use_DSA **model_coef \(~\,~\,\) P1,P2,aP,omP,nP,ta0

Available coefficients

In summary, the following coefficients are available:

R0

yield radius in the nonlinear isotropic hardening term at slip system scale.

K, n

viscoplastic Norton law coefficients for \(\dot{\lambda} = \left<f/K\right>^n\).

c, d

nonlinear Armstrong-Frederick kinematic hardening.

a00, a90, ash, ap45, an45

initial back stresses coefficients.

f0, D1, s1, aL, aT, aN, fN, eN, sN, fu

Damage (**use_Rdamage).

f02, D12, t12, fN2, eN2, sN2

Secondary voids (**use_void_meso).

c3, r3, Q3, b3, p3, gvC

Mohr Coulomb model (**use_MC).

P1, P2, aP, omP, nP, ta0

DSA model (**use_DSA).

Note

There is an example file in the test database directory ZebFront_test/INP named *.inp

The support document is…