Nonlocal formulation - TrabBoneNL3d

The model is regularized by the over-nonlocal formulation with damage driven by a combination of local and nonlocal cumulated plastic strain

$$\hat{\kappa} = (1-m)\kappa + m\bar{\kappa},$$ (219)

where $m$ is a dimensionless material parameter (typically $m>1$) and

$$\bar{\kappa}(x) = \int\limits_V \alpha(x,s)\kappa(s)\,{\rm d}s$$ (220)

is the nonlocal cumulated plastic strain. The nonlocal weight function is defined as

$$\alpha(x,s) = \frac{\alpha_0(\Vert x-s\Vert )}{\int\limits_V\alpha_0(\Vert x-t\Vert )\,{\rm d}t}$$ (221)

where

$$\alpha_0(r) =\left\{\begin{array}{cc} \left(1-\frac{r^2}{R^2}\right)^2 & \mbox{ if } r\le R \\ 0 & \mbox{ if } r> R \end{array}\right.$$ (222)

Parameter $R$ is related to the internal length of the material. The model description and parameters are summarized in Tab. 47.

Table 47: Nonlocal formulation of anisotropic elastoplastic model with isotropic damage - summary.

Description	Nonlocal anisotropic elastoplastic model with isotropic damage
Record Format	TrabBoneNL3d (in) # d(rn) # eps0(rn) # nu0(rn) # mu0(rn) # expk(rn) # expl(rn) # m1(rn) # m2(rn) # rho(rn) # sig0pos(rn) # sig0neg(rn) # chi0pos(rn) # chi0neg(rn) # tau0(rn) # plashardfactor(rn) # expplashard(rn) # expdam(rn) # critdam(rn) # m(rn) # R(rn) #
Parameters	- material number
	- d material density
	- eps0 Young modulus (at zero porosity)
	- nu0 Poisson ratio (at zero porosity)
	- mu0 shear modulus (at zero porosity)
	- m1 first eigenvalue of the fabric tensor
	- m2 second eigenvalue of the fabric tensor
	- rho volume fraction of the solid phase
	- sig0pos yield stress in tension
	- tau0 yield stress in shear
	- chi0pos interaction coefficient in tension
	- chi0neg interaction coefficient in compression
	- plashardfactor hardening parameter
	- expplashard exponent in the hardening law
	- expdam exponent in the damage law
	- critdam critical damage
	- expk exponent $k$ in the expression for elastic stiffness
	- expl exponent $l$ in the expression for elastic stiffness
	- expq exponent $q$ in the expression for tensor $F$
	- expp exponent $p$ in the expression for tensor $F$
	- m over-nonlocal parameter
	- R nonlocal interaction radius
Supported modes	3dMat