Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage

Abstract

The paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials, i.e., materials for which the yield limits soften as a function of the total strain. Both plasticity and damage models are considered. It is verified analytically in a simple way that the nonlocal integral-type model with degrading yield limit depending on the total strain works correctly if and only one adopts an unconventional nonlocal formulation introduced in 1994 by Vermeer and Brinkgreve (and in 1996 by Planas, and by Stro¨mberg and Ristinmaa), which is here called, for the sake of brevity, over-nonlocal because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is compensated by assigning to the nonlocal variable weight greater than 1 (this is equivalent to a nonlocal variable with a smooth positive weight function of total weight greater than 1, normalized by superposing a negative delta-function spike at the center). The spectral approach readily confirms that the nonlocal integral-type generalization of softening plasticity with an additive format gives correct localization properties only if an over-nonlocal formulation is adopted. By contrast, the nonlocal integral-type generalization of softening plasticity with a multiplicative format provides realistic localization behavior, just like the nonlocal integral-type damage model, and thus does not necessitate an over-nonlocal formulation. The localization behavior of explicit and implicit gradient-type models is also analyzed. A simple analysis shows that plasticity and damage models with gradient-type localization limiter, whether explicit or implicit, have very different localization behaviors

Spectral Localization Analysis in Nonlocal Softening Materials

“…1, and m is an empirical coefficient (over-nonlocal parameter) greater than 1 has showed by many authors [14]. The refinement of Eq.…”

“…In this paper some of the main results presented in [14] are reported. Firstly, the basic concept of nonlocal formulation of integraltype is presented with the relation between it and the gradient enhancements.…”

“…To improve the performance in large post-peak deformations, Vermeer and Brinkgreve [8] proposed a novel formulation in which a nonlocal averaging integral with a weight function everywhere positive is modified by multiplying the weight function by a factor greater than 1 and compensating for it by adding at the center a negative deltafunction spike, equivalent to subtracting the local variable (this is unlike applying the common rule of mixture because the nonlocal component in the binary mixture of the local and nonlocal variables has a weight greater than 1, compensated by a negative weight for the other component). This formulation was termed overnonlocal in [14]. Another well-known regularizing technique is based on an explicit secondorder gradient model, which is weakly nonlocal [3,[9][10][11][12].…”

A one-dimensional dynamic analysis of strain-gradient viscoplasticity

“…The ill-posedness of BVP is characterized by the loss of ellipticity in statics and of hyperbolicity in dynamics when materials soften. In dynamics, the differential equations of motion become ill-posed as the wave velocity becomes imaginary (Luzio(2005)). A possibility to overcome such the mathematical difficulty is to introduce the gradient of plastic strain or the damage parameter fields in order to penalize possible sharp localization Q.S.…”

“…The regularisation techniques of nonlocal or by gradient approaches for damage models were developed by many authors, i.e. Peerlings et al (1996Peerlings et al ( , 1998Peerlings et al ( , 2001), Askes et al(2000Askes et al( , 2002aAskes et al( , 2002b, Luzio and Bazant (2005). An analytical solution of dynamic behaviour for a micro beam is reported in Kong et al (2009).…”