The analysis of stability and bifurcation is studied in nonlinear mechanics with dissipative mechanisms: plasticity, damage or fracture. The description of the behaviour is based on definittion of a set of internal variables. This framework allows a unified description of the material behaviour via two potentials: the free energy (for reversible part) and the potential of dissipation to describe the irreversibility. In the case of standard generalized materials the internal evolution is governed by variational inequalities which depend on the mechanism of dissipation. These inequalities are obtained under energetic considerations in an unified description based upon energy and driving forces associated to internal parameters evolution. Criteria for existence and uniqueness of the system evolution are then deduced. Examples are presented for plasticity, fracture, delamination and damaged materials, on specific geometry and external loading parameters.

The buckling phenomenon represents one of the main failure modes observed in pressure equipments, due to the thinness of the shell structures involved and the compressive stresses stemming from standard loads such as external pressure. Some methodologies exist for the buckling design of pressure vessels, assembled in codes such as the ASME Boiler and Pressure Vessel Code (BPVC) or the French CODAP. These rules are either analytical but limited to a certain range of validity, or in the form of recommendations for performing finite element analyses.

In view of this, the objective of the present study is to provide a purely analytical methodology for the elastoplastic buckling analysis of a complete equipment under external pressure, made up of a cylindrical shell and semi-ellipsoidal ends, allowing thus for a reliable and efficient design. First, original closed-form solutions are obtained for the two separate parts. Corrections are made so as to take account of the connection between the assembled components. Then, different buckling scenarii can be envisaged. The buckling response of the pressure equipment may initially involve either the cylindrical shell or the ends, depending on the respective critical pressures. In both cases, buckling may be elastic, plastic or occur intermediately at the threshold pressure, within the so-called continuum of bifurcation points. An analytical tool is finally built, which enables to determine the buckling type but also the critical pressure expected in practice, after comparing all the possible scenarii. This predictive tool makes it possible to obtain very efficiently mappings of the buckling types and critical pressures by varying any geometric and/or material parameters, for optimization purposes. It is first validated by comparison with numerical finite element computations carried out on complete equipments, and then confronted to the results obtained with the use of the French CODAP recommendations.

The proper functionality of a human body relies on several continuous physical processes, many of which are carried out through biological ducts/tubes. For instance, veins, arteries and airways are natural conduit systems where blood and air are conveyed towards them respectively into the human body. Previous studies have shown that those tubular components are prone to collapse under critical conditions of internal and external pressure, resulting in malfunctioning of main physical processes. This coupled problem of flow through collapsible biological tubes has been studied for several decades through extensive numerical and experimental work. In the present study, the actual coupled fluid-structure problem is simplified by examining the response of elastic tubes prone to collapse under uniform external pressure, emphasizing on nonlinear material behavior. The problem is approached numerically using Abaqus/Standard for analyzing tube models with values of D/t ratio ranging from 12 to 30 and considering different nonlinear elastic material properties. Material behavior is described using (a) a power-law constitutive model that follows deformation theory of plasticity and (b) hyperelastic models. The main purpose of the paper is to examine whether a small deviations from linear elastic behavior is able to cause a localized collapse pattern and trigger buckle propagation. Elastic tubes with biological material properties are used. Results from three dimensional (3D) and two-dimensional (2D) models are obtained and compared. The numerical results show that localized collapse and propagation is possible to occur in tubes made of nonlinear elastic material.

Steel plates are employed in many structural forms such as the webs and flanges of beams and columns, rack sections and bridge girders. These plated structures are often perforated for ease of manufacture (e.g. cope holes), service (e.g. passage of pipes) or for aesthetic purposes (e.g. cladding). The addition of perforations can also be structurally beneficial: reducing structural mass without greatly compromising the elastic buckling capacity. Depending on the orientation, size, and location of a regular array of perforations, the elastic buckling capacity may even increase.

In this paper, an algorithm is used to determine what arrangement of perforations is more likely to benefit the elastic buckling capacity of a given slender plate. This geometry is assessed computationally, and a shape grammar approach is used to optimise the structural weight to elastic buckling capacity ratio.