A stochastic nonlocal damage model

Deterministic nonlocal damage models permit avoiding spurious mesh sensitivity and predict structure size effect which is in accordance with experimental observations on notched specimens made of concrete. However, these deterministic models are unable to predict volume size effect exhibited by unnotched specimens in direct tension because the effect of heterogeneity has been introduced via a deterministic strain-softening behaviour.The main idea of our model is to consider two scales: the meso-scale (the representative volume) of size l ^r and the micro-scale, of size d (d l ^r). The plane is subdivided in square cells of size d. Each cell is supposed to be homogeneous and behaves in a purely brittle manner. The heterogeneity of the material properties are introduced by attributing to each cell a random threshold value picked according to a power law distribution function of parameter m. The breaking criterion is nonlocal.In finite element calculations, each cell is discretized in finite element meshes which have the same threshold value. Each finite element mesh behaves in a purely brittle manner and its breaking criterion is the ratio of the volume average of the elastic energy over a cube of size l ^r and the threshold value.Numerical simulations on notched and unnotched specimens will show that the proposed model yields mesh insensitive calculations and predicts both volume and structure size effects which are qualitatively in accordance with experimental observations.     

Reaction of Lawesson's reagent with Mannich base hydrochlorides: synthesis of novel 4H-1,3,2-oxathiaphosphorine-2-sulfide derivatives

In a simple and general protocol, the treatment of Mannich base hydrochlorides, derived from acyclic and cyclic ketones, with Lawesson's reagent, leads to novel 4H-1,3,2-oxathiaphosphorine derivatives in good yields. A possible reaction mechanism, involving a [4+2] cycloaddition, is proposed.