Plastic models

In PeriLab J2 plasticity model (von Mises plasticity) is implemented for correspondence-based and ordinary state-based Peridynamics. The plastic model was taken from Peridigm and translated into julia. The theoretical basis can be found here LammiC2014.

!!! info "Isotropic model" This model works only for isotropic yield stresses.

Fundamental Theory

J2 Plasticity (von Mises Plasticity)

The J2 plasticity theory is based on the von Mises yield criterion, which states that yielding occurs when the von Mises stress reaches a critical value (yield stress). This theory is particularly suitable for ductile materials like metals.

Stress Decomposition

The total stress tensor σ is decomposed into:

$$\boldsymbol{\sigma} = \boldsymbol{s} + p\mathbf{I}$$

where:

• s is the deviatoric stress tensor
• p is the spherical (hydrostatic) stress
• I is the identity tensor

The spherical stress is calculated as:

$$p = \frac{1}{3}\text{tr}(\boldsymbol{\sigma}) = \frac{1}{3}(\sigma_{11} + \sigma_{22} + \sigma_{33})$$

The deviatoric stress tensor is:

$$\boldsymbol{s} = \boldsymbol{\sigma} - p\mathbf{I}$$

von Mises Stress

The von Mises equivalent stress is defined as:

$$\sigma_{vM} = \sqrt{\frac{3}{2}\boldsymbol{s}:\boldsymbol{s}} = \sqrt{\frac{3}{2}s_{ij}s_{ij}}$$

This can also be written as:

$$\sigma_{vM} = \sqrt{3J_2}$$

where J₂ is the second invariant of the deviatoric stress tensor.

Yield Criterion

The yield condition is:

$$f = \sigma_{vM} - \sigma_y \leq 0$$

where:

• f is the yield function
• σᵧ is the yield stress (possibly reduced by flaw functions)

Elastic regime: If f < 0, the material remains elastic and stresses are unchanged.

Plastic regime: If f ≥ 0, plastic deformation occurs and stresses must be returned to the yield surface.

Return Mapping Algorithm

When the trial stress exceeds the yield surface, a radial return mapping is applied to the deviatoric stress:

Step 1: Calculate Trial Deviatoric Stress Magnitude

$$|\boldsymbol{s}^{trial}| = \frac{\sigma_{vM}^{trial}}{\sqrt{\frac{2}{3}}}$$

Step 2: Scale Deviatoric Stress to Yield Surface

$$\boldsymbol{s}^{t+\Delta t} = \boldsymbol{s}^{trial} \cdot \frac{\sqrt{\frac{2}{3}} \cdot \sigma_y}{|\boldsymbol{s}^{trial}|}$$

Step 3: Reconstruct Total Stress

$$\boldsymbol{\sigma}^{t+\Delta t} = \boldsymbol{s}^{t+\Delta t} + p^{t+\Delta t}\mathbf{I}$$

Note: The spherical stress remains unchanged during plastic return, as J2 plasticity assumes plastic incompressibility (volume preservation).

Equivalent Plastic Strain Update

The equivalent plastic strain is updated using an incremental approach that is independent of the specific return mapping algorithm:

Deviatoric Strain Increment

The deviatoric part of the strain increment is:

$$\boldsymbol{e}^{dev} = \boldsymbol{\varepsilon}^{inc} - \frac{1}{3}\text{tr}(\boldsymbol{\varepsilon}^{inc})\mathbf{I}$$

Plastic Strain Increment Calculation

The plastic strain increment is computed by projecting onto the flow direction:

$$\mathbf{A} = \boldsymbol{e}^{dev} - \frac{\boldsymbol{s}^{t+\Delta t} - \boldsymbol{s}^t}{2G}$$

$$\mathbf{B} = \frac{1}{2}\left(\frac{\boldsymbol{s}^{t+\Delta t}}{|\boldsymbol{s}^{t+\Delta t}|} + \frac{\boldsymbol{s}^t}{|\boldsymbol{s}^t|}\right)$$

$$\Delta \varepsilon_p^{eq} = \max\left(0, \sqrt{\frac{2}{3}} , \mathbf{A}:\mathbf{B}\right)$$

where:

• G is the shear modulus
• The contraction A:B represents the tensor inner product

Total Equivalent Plastic Strain

$$\varepsilon_p^{eq,t+\Delta t} = \varepsilon_p^{eq,t} + \Delta \varepsilon_p^{eq}$$

Material Parameters

The model requires the following material parameters:

1. Shear Modulus (G): Controls elastic shear response and plastic strain increment calculation
2. Yield Stress (σᵧ): Critical stress at which plastic deformation begins

Flaw Function

The yield stress can be spatially reduced using a flaw function:

$$\sigma_y^{reduced} = f_{flaw}(\mathbf{x}) \cdot \sigma_y$$

This allows modeling of material defects, damage, or heterogeneity in a simple way.