The Stochastic Wave Equation with Fractional Noise: a random field approach

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in \cite{dalang99}, when the noise is white in time. Under this condition, we show that the solution is $L^2(\Omega)$-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is {\em different} (and more general) than the one obtained for the wave equation

The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution

In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in (0,T) \times \bR^d, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$

On the numerical implementation of elasto-plastic constitutive equations for metal forming

National audienceThis paper is devoted to the time integration of elasto-plastic constitutive models, in view of their implementation in finite element software for the simulation of metal forming processes. Both implicit and explicit time integration schemes are reviewed and presented in algorithmic form. The incremental kinematics are also treated, so that the proposed algorithms can be used stand-alone, outside a finite element code, or they can serve to implement non-classical incremental kinematics. Full algorithms are provided, along with examples of application to non-monotonic loading for a mild steel and a dual phase steel

Overview of the theoretical relations between necking and strain localization criteria.

Many criteria have been developed during last decades to predict diffuse or localized necking and shear banding. The lack of confrontation of these models with each other on relevant applications makes their choice difficult for the designer. It is proposed to reformulate these plastic instability criteria in an unified framework, to compare their theoretical bases to establish links between them and then to highlighten their limitations. In the case of diffuse necking, a comparison is made between the criteria based on bifurcation analysis and on those based on maximum force principle for elastic-plastic materials. In the case of localized modes, it is shown that the predictions of the Marciniak – Kuczynski approach, based on a multizone model, tend to those of the loss of ellipticity criterion when the initial defect size tends to zero (no initial defect introduced). In the case of elasto-viscoplastic behavior, an approach based on a linear stability analysis is mentioned

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise $u_t={1/2}\Delta u+ u \diamond \dot{W}$ in \bR_{+} \times \bR^d, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order $\alpha<d$, then the sufficient condition for the existence of the solution is $d \leq 2+\alpha$ (if $H>1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of $k$ independent $d$-dimensional Brownian motions

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

http://link.springer.com/article/10.1007%2Fs00419-012-0620-xThis paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the "thickness" direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications – both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems

Elastic-plastic ductile damage model based on strain-rate plastic potential

Modeling of ductile damage is generally done using analytical potentials, which are expressed in the stress space. In this paper, for the first time it is shown that strain rate potentials which are exact conjugate of the stress-based potentials can be instead used to model the dilatational response of porous polycrystals. A new integration algorithm is also developed. It is to be noted that a strain-rate based formulation is most appropriate when the plastic flow of the matrix is described by a criterion that involves dependence on all stress invariants. In such cases, although a strain-rate potential is known, the stress-based potential cannot be obtained explicitly. While the proposed framework based on strain-rate potentials is general, for comparison purposes in this work we present an illustration of the approach for the case of a porous solid with von Mises matrix containing randomly distributed spherical cavities. Comparison between simulations using the strain-rate based approach and the classical stress-based Gurson’s criterion in uniaxial tension is presented. These results show that the model based on a strain-rate potential predicts the dilatational response with the same level of accuracy

Numerical Investigation of the Limit Strains in Sheet Forming Involving Bending

In this work, the finite element method is used to simulate a typical FLD test over tools of different radii. Parameters like the mesh density, element type, numerical determination of the onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit strain for plane strain tension has been determined as a function of the thickness vs. tool radius (t/R) ratio. These simulations confirm that increasing the curvature of the tool increases the value of the limit strains. They also reveal that, as soon as bending becomes important, the practical relevance of the limit strains diminishes - At least with their current definition. The need for new strain localization models is emphasized, together with some of the associated challenges.Projet ANR FORME

Numerical Investigation of the Limit Strains in Sheet Forming Involving Bending

In this work, the finite element method is used to simulate a typical FLD test over tools of different radii. Parameters like the mesh density, element type, numerical determination of the onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit strain for plane strain tension has been determined as a function of the thickness vs. tool radius (t/R) ratio. These simulations confirm that increasing the curvature of the tool increases the value of the limit strains. They also reveal that, as soon as bending becomes important, the practical relevance of the limit strains diminishes - At least with their current definition. The need for new strain localization models is emphasized, together with some of the associated challenges.Projet ANR FORME

On the numerical implementation of elasto-plastic constitutive equations for metal forming

This paper is devoted to the time integration of elasto-plastic constitutive models, in view of their implementation in finite element software for the simulation of metal forming processes. Both implicit and explicit time integration schemes are reviewed and presented in algorithmic form. The incremental kinematics are also treated, so that the proposed algorithms can be used stand-alone, outside a finite element code, or they can serve to implement non-classical incremental kinematics. Full algorithms are provided, along with examples of application to non-monotonic loading for a mild steel and a dual phase steel