Nonlocal isotropic damage model for tensile failure - Idmnl1

Nonlocal version of isotropic damage model from Section 1.5.6. The nonlocal averaging acts as a powerful localization limiter. In the standard version of the model, damage is driven by the nonlocal equivalent strain $\bar{\varepsilon}$, defined as a weighted average of the local equivalent strain:

$$\bar{\varepsilon}($$$x$$$) = \int_V\alpha($$$x$$$,$$$\xi$$$)\tilde\varepsilon($$$\xi$$$)\;{\rm d}$$$\xi$

In the “undernonlocal” formulation, the damage-driving variable is a combination of local and nonlocal equivalent strain, $m\bar{\varepsilon}+(1-m)\tilde\varepsilon$, where $m$ is a parameter between 0 and 1. (If $m>1$, the formulation is called “overnonlocal”; this case is useful for nonlocal plasticity but not for nonlocal damage.)

Instead of averaging the equivalent strain, one can average the compliance variable $\gamma$, directly related to damage according to the formula $\gamma=\omega/(1-\omega)$.

The weight function $\alpha$ contains a certain parameter with the dimension of length, which is in general called the characteristic length. Its specific meaning depends on the type of weight function. The following functions are currently supported:

• Truncated quartic spline, also called the bell-shaped function,
  $$\alpha_0(s) = \left\langle 1-\frac{s^2}{R^2}\right\rangle^2$$
  where $R$ is the interaction radius (characteristic length) and $s$ is the distance between the interacting points. This function is exactly zero for $s\ge R$, i.e., it has a bounded support.
• Gaussian function
  $$\alpha_0(s) = \exp\left(-\frac{s^2}{R^2}\right)$$
  which is theoretically nonzero for an arbitrary large $s$ and thus has an unbounded support. However, in the numerical implementation the value of $\alpha_0$ is considered as zero for $s>2.5R$.
• Exponential function
  $$\alpha_0(s) = \exp\left(-\frac{s}{R}\right)$$
  which also has an unbounded support, but is considered as zero for $s>6R$. This function is sometimes called the Green function, because in 1D it corresponds to the Green function of the Helmholtz-like equation used by implicit gradient approaches.
• Piecewise constant function
  $$\alpha_0(s) =\left\{\begin{array}{cc} 1 & \mbox{ if } s\le R \\ 0 & \mbox{ if } s> R \end{array}\right.$$
  which corresponds to uniform averaging over a segment, disc or ball of radius $R$.
• Function that is constant over the finite element in which point $x$ is located, and is zero everywhere else. Of course, this is not a physically objective definition of nonlocal averaging, since it depends on the discretization. However, this kind of averaging was proposed in a boundary layer by Prof. Bazant and was implemented into OOFEM for testing purposes.
• Special function
  $$\alpha_0(s) =\int_{-\infty}^\infty \exp\left(-\frac{\sqrt{s^2+t^2}}{R}\right)$$d$$t$$
  obtained by reduction of the exponential function from 2D to 1D. The integral cannot be evaluated in closed form and is computed by OOFEM numerically. This function can be used in one-dimensional simulations of a two-dimensional specimen under uniaxial tension; for more details see [11].

The above functions depend only on the distance $s$ between the interacting points and are not normalized. If the normalizing condition

$$\int_{V_\infty} \alpha($$$x$$$,$$$\xi$$$)\;{\rm d}$$$\xi$$$= 1$$

is imposed in an infinite body $V_{\infty}$, it is sufficient to scale $\alpha_0$ by a constant and set

$$\alpha($$$x$$$,$$$\xi$$$)=\frac{\alpha_0(\Vert\mbox{\boldmath $x$}-\mbox{\boldmath $\xi$}\Vert)}{V_{r\infty}}$$

where

$$V_{r\infty} = \int_{V_\infty} \alpha_0(\Vert$$$\xi$$$\Vert)\;{\rm d}$$$\xi$$$$$

Constant $V_{r\infty}$ can be computed analytically depending on the specific type of weight function and the number of spatial dimensions in which the analysis is performed. Since the factor $1/V_{r\infty}$ can be incorporated directly in the definition of $\alpha_0$, this case is referred to as “no scaling”.

If the body of interest is finite (or even semi-infinite), the averaging integral can be performed only over the domain filled by the body, and the volume contributing to the nonlocal average at a point $x$ near the boundary is reduced as compared to points $x$ far from the boundary or in an infinite body. To make sure that the normalizing condition

$$\int_{V} \alpha($$$x$$$,$$$\xi$$$)\;{\rm d}$$$\xi$$$= 1$$

holds for the specific domain $V$, different approaches can be used. The standard approach defines the nonlocal weight function as

$$\alpha($$$x$$$,$$$\xi$$$)=\frac{\alpha_0(\Vert\mbox{\boldmath $x$}-\mbox{\boldmath $\xi$}\Vert)}{V_r(\mbox{\boldmath $x$})}$$

where

$$V_r($$$x$$$) = \int_{V} \alpha_0(\Vert$$$x$$$-$$$\xi$$$\Vert)\;{\rm d}$$$\xi$$$$$

According to the approach suggested by Borino, the weight function is defined as

$$\alpha($$$x$$$,$$$\xi$$$)=\frac{\alpha_0(\Vert\mbox{\boldmath $x$}-\mbox{\boldmath $\xi$}... ... $x$})}{V_{r\infty}}\right)\delta(\mbox{\boldmath $x$}-\mbox{\boldmath $\xi$})$$

where $\delta$ is the Dirac distribution. One can also say that the nonlocal variable is evaluated as

$$\bar{\varepsilon}($$$x$$$) = \frac{1}{V_{r\infty}}\int_V\alpha_0(\Vert$$$x$$$-$$$\xi$$$\Vert)\tilde\epsilon($$$\xi$$$)\;{\rm d}$$$\xi$$$+\left(1-\frac{V_r(\mbox{\boldmath $x$})}{V_{r\infty}}\right)\tilde\varepsilon(\mbox{\boldmath $x$})$$

The term on the right-hand side after the integral is a multiple of the local variable, and so it can be referred to as the local complement.

In a recent paper [11], special techniques that modify the averaging procedure based on the distance from a physical boundary of the domain or on the stress state have been considered. The details are explained in [11]. These techniques can be invoked by setting the optional parameter nlVariation to 1, 2 or 3 and specifying additional parameters $\beta$ and $\zeta$ for distance-based averaging, or $\beta$ for stress-based averaging.

The model parameters are summarized in Tabs. 24 and 25.

Table 24: Nonlocal isotropic damage model for tensile failure - summary.

Description

Nonlocal isotropic damage model for concrete in tension

Record Format

Idmnl1 (in) # d(rn) # E(rn) # n(rn) # [tAlpha(rn) #] [equivstraintype(in) #] [k(rn) #] [damlaw(in) #] e0(rn) # [ef(rn) #] [At(rn) #] [Bt(rn) #] [md(rn) #] r(rn) # [regionMap(ia) #] [wft(in) #] [averagingType(in) #] [m(rn) #] [scalingType(in) #] [averagedQuantity(in) #] [nlVariation(in) #] [beta(rn) #] [zeta(rn) #] [maxOmega(rn) #]

Parameters

- material number

- d material density

- E Young's modulus

- n Poisson's ratio

- tAlpha thermal expansion coefficient

- equivstraintype allows to choose from different definitions of equivalent strain, same as for the local model; see Tab. 23

- k ratio between uniaxial compressive and tensile strength, needed only if equivstraintype=3, default value 1

- damlaw allows to choose from different damage laws, same as for the local model; see Tab. 23 (note that parameter wf cannot be used for the nonlocal model)

- e0 strain at peak stress (for damage laws 0,1,2,3), limit elastic strain (for damage law 4), characteristic strain (for damage law 5)

- ef strain parameter controling ductility, has the meaning of strain (for damage laws 0 and 1), the tangent modulus just after the peak is $E_t=-f_t/(\varepsilon_f-\varepsilon_0)$

- At parameter of Mazars damage law, used only by law 4

- Bt parameter of Mazars damage law, used only by law 4

- md exponent, used only by damage law 5, default value 1

- r nonlocal characteristic length $R$; its meaning depends on the type of weight function (e.g., interaction radius for the quartic spline)

- regionMap map indicating the regions (currently region is characterized by cross section number) to skip for nonlocal avaraging. The elements and corresponding IP are not taken into account in nonlocal averaging process if corresponding regionMap value is nonzero.

- wft selects the type of nonlocal weight function:

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)

-- continued in Tab. 25 --

Table 25: Nonlocal isotropic damage model for tensile failure - continued.

Description

Nonlocal isotropic damage model for concrete in tension

- averagingType activates a special averaging procedure, default value 0 does not change anything, value 1 means averaging over one finite element (equivalent to wft=5, but kept here for compatibility with previous version)

- m multiplier for overnonlocal or undernonlocal formulation, which use m-times the local variable plus $(1-m)$-times the nonlocal variable, default value 1

- scalingType selects the type of scaling of the weight function (e.g. near a boundary):

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)

- averagedQuantity selects the variable to be averaged, default value 1 corresponds to equivalent strain, value 2 activates averaging of compliance variable

- nlVariation activates a special averaging procedure, default value 0 does not change anything, value 1 means distance-based averaging (the characteristic length is linearly reduced near a physical boundary), value 2 means stress-based averaging (the averaging is anisotropic and the characteristic length is affected by the stress), value 3 means distance-based averaging (the characteristic length is exponentially reduced near a physical boundary)

- beta parameter $\beta$, required only for distance-based and stress-based averaging (i.e., for nlVariation=1, 2 or 3)

- zeta parameter $\zeta$, required only for distance-based averaging (i.e., for nlVariation=1 or 3)

- maxOmega maximum damage, used for convergence improvement (its value is between 0 and 0.999999 (default), and it affects only the secant stiffness but not the stress)

Supported modes

3dMat, PlaneStress, PlaneStrain, 1dMat

Features

Adaptivity support