Implicit gradient formulation

The gradient formulation can be conceived as the differential counterpart to the integral formulation. The nonlocal cumulated plastic strain is computed from a Helmholtz-type differential equation

$\displaystyle \bar{\kappa} - l^2\nabla^2\bar{\kappa} = \kappa$ (51)

with homogeneous Neumann boundary condition

$\displaystyle \frac{\partial\bar{\kappa}}{\partial n} = 0.$ (52)

In (51), $ l$ is the length scale parameter and $ \nabla$ is the Laplace operator.

The model description and parameters are summarized in Tabs. 11 and 12. Note that the internal length parameter r has the meaning of the radius of interaction $ R$ for the integral version (and thus has the dimension of length) but for the gradient version it has the meaning of the coefficient $ l^2$ multiplying the Laplacean in (51), and thus has the dimension of length squared.

Table 11: Nonlocal integral Mises plasticity - summary.
Description Nonlocal Mises plasticity with isotropic hardening
Record Format MisesMatNl (in) # d(rn) # E(rn) # n(rn) # sig0(rn) # H(rn) # omega_crit(rn) #a(rn) #r(rn) #m(rn) #[wft(in) #][scalingType(in) #]
Parameters - material number
  - d material density
  - E Young's modulus
  - n Poisson's ratio
  - sig0 initial yield stress in uniaxial tension (compression)
  - H hardening modulus
  - omega_crit critical damage
  - a exponent in damage law
  - r nonlocal interaction radius $ R$ from eq. (50)
  - m over-nonlocal parameter
  - wft selects the type of nonlocal weight function (see Section 1.5.7):
1 -
default, quartic spline (bell-shaped function with bounded support)
2 -
Gaussian function
3 -
exponential function (Green function in 1D)
4 -
uniform averaging up to distance $ R$
5 -
uniform averaging over one finite element
6 -
special function obtained by reducing the 2D exponential function to 1D (by numerical integration)
  - scalingType selects the type of scaling of the weight function (e.g. near a boundary; see Section 1.5.7):
1 -
default, standard scaling with integral of weight function in the denominator
2 -
no scaling (the weight function normalized in an infinite body is used even near a boundary)
3 -
Borino scaling (local complement)
Supported modes 1dMat, PlaneStrain, 3dMat



Table 12: Gradient-enhanced Mises plasticity - summary.
Description Gradient-enhanced Mises plasticity with isotropic damage
Record Format MisesMatGrad (in) # d(rn) # E(rn) # n(rn) # sig0(rn) # H(rn) # omega_crit(rn) #a(rn) #r(rn) #m(rn) #
Parameters - material number
  - d material density
  - E Young's modulus
  - n Poisson's ratio
  - sig0 initial yield stress in uniaxial tension (compression)
  - H hardening modulus
  - omega_crit critical damage
  - a exponent in damage law
  - r internal length scale parameter $ l^2$ from eq. (51)
  - m over-nonlocal parameter
Supported modes 1dMat, PlaneStrain, 3dMat


Borek Patzak
2018-01-02