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

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

with homogeneous Neumann boundary condition

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