J2 Plasticity

J2 Plasticity#

../../../../_images/j2-mises.png

Tension test of a coupon using the \(J_2\) plasticity model from the STAIRLab gallery.#

J2Plasticity is a multi dimensional material model that incorporates plasticity using the von Mises \(J_2\) yield criterion, with nonlinear isotropic hardening.

Model.nDMaterial("J2", tag, K, G, Fy, Fs, Hsat, Hiso)
Parameters:

  • tag (integer) – unique tag identifying material

  • K (float) – Bulk modulus, \(\kappa\) [1]

  • G (float) – Shear modulus, \(\mu\) [1]

  • Fy (float) – Initial yield stress, \(F_y\) [1]

  • Fs (float) – Saturation yield stress

  • Hsat (float) – exponential hardening parameter

  • Hiso (float) – linear isotropic hardening modulus

Notes#

Parameters#

  • K - Bulk modulus \(\kappa\)

  • G - Shear modulus \(\mu\)

  • E - Young’s modulus \(E\)

    Note

    Updates to \(E\) are performed at constant Poisson ratio \(\nu\).

  • Fy - Initial yield stress \(F_y\)

Examples#

nDMaterial J2 [incr i] -E $E -G $G $Fy $Fs $Hsat $Hiso $eta
nDMaterial J2 [incr i] -E $E -G $G $Fy $Fs $Hsat $Hiso $eta -density $density
nDMaterial J2 [incr i] -E $E -nu $nu $Fy $Fs $Hsat $Hiso $eta -density $density

Theory#

In the elastic range, the material response follows an Elastic Isotropic formulation:

\[\boldsymbol{T} = K \operatorname{tr} \boldsymbol{E}_e + 2 G \operatorname{dev} \boldsymbol{E}_e\]

Plastic response is distinguished by the yield function \(f\)

\[f (\boldsymbol{T},q) \triangleq \| \operatorname{dev} \boldsymbol{T} \| - \sqrt{\tfrac{2}{3}} \, q^{\mathrm{iso}}(\bar{\epsilon}_{\mathrm{p}})\]

where \(\bar{\epsilon}_{\mathrm{p}}\) is the scalar equivalent plastic tensile strain, and \(q^{\mathrm{iso}}\) is a scalar function that defines the saturation isotropic hardening given by:

\[q^{\mathrm{iso}}(\bar{\epsilon}_{\mathrm{p}}) = H_{\mathrm{iso}} \bar{\epsilon}_{\mathrm{p}} + F_{s} + (F_y - F_{s}) \exp \left(-H_{\mathrm{s}} \bar{\epsilon}_{\mathrm{p}} \right)\]

Note

This is identical to the hardening function for Drucker Prager, when \(F_y \equiv F_0\).

This hardening rule is equivalent to the model implemented by FEAP. The flow rules are

\[\dot{\boldsymbol{E}}_{\mathrm{p}} = \gamma \frac{\partial f}{\partial \boldsymbol{T}}\]

where \(\gamma\) is the plastic consistency parameter and \(\boldsymbol{E}_{\mathrm{p}}\) is the plastic strain tensor. linear viscosity is exhibited with \(\gamma = \frac{\phi}{\eta}\) ( if \(\phi > 0\) )

Backward Euler integration is employed in the implementation.

Note

  • For linear isotropic hardening, set \(F_{\infty} = F_0\)

  • For rate independent cases, set \(\eta = 0\).

References#

Code Developed by: Ed Love

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