Advances in Direct Methods for Materials and Structures

There exist a number of significant problems where the assumptions of limit and shakedown analysis, i.e. the bounding theorems, are not fully satisfied. Principal amongst such problems are those where the yield surface is convex but the flow rule is non-associated. This includes limit states in geomechanics where yield is pressure dependent but flow remains volume conserving. Coulomb friction between elastic bodies shows related behaviour. The paper explores the extent to which the classical limit theorems may be extended to the Drucker-Prager yield condition with a non-associated flow rule where the plastic strain rate involves no volume change. Bounds that correspond to the classical kinematic and static bounds are derived which defines a range within which consistent limit state solutions will exist, i.e. the limit state is not generally unique.

Shape Memory Alloys (SMAs) belong to the class of so-called smart materials that offer promising perspectives in various fields such as aeronautics, robotics, biomedicals or civil engineering. For elastic-plastic materials, there is an established correlation between fatigue and energy dissipation. In particular, high-cycle fatigue occurs when the energy dissipation remains bounded in time. Although the physical mechanisms in SMAs differ from plasticity, the hysteresis that is commonly observed in the stress-strain response of those materials shows that some energy dissipation occurs. It can be reasonably assumed that situations where the energy dissipation remains bounded are the most favorable for fatigue durability. In this contribution, we present a direct method for determining if the energy dissipation in a SMA structure (submitted to a prescribed loading history) is bounded or not. That method is direct in the sense that nonlinear incremental analysis is completely bypassed. The proposed method rests on a suitable extension of the well-known Melan theorem. An application related to biomedical stents is presented to illustrate the method.

It is well known that in high cycle fatigue (HCF), macroscopically, structures undergo elastic shakedown and the stress level commonly determines the lifetime. In this domain, the fatigue phenomena is due to local plasticity at the grain scale. Therefore, some multiscale HCF multiaxial fatigue criteria were proposed, among them the well-known Dang Van criterion. This criterion supposes that in a polycrystal, some misoriented grains can undergo plastic shakedown which conducts to crack initiation. The objective of this work is to validate this assumption by conducting numerical simulations on polycrystalline aggregates. As it is necessary to estimate the stabilized state in each grain of the polycrystal, classical incremental simulations are not the best way as it will be highly time-consuming because of the size of the aggregate. In the recent years, Pommier proposed a method called Direct Cyclic Algorithm to obtain the stabilized response of a structure under cyclic periodic loading, which it is shown to be more efficient compared to an incremental analysis in such situation. However, errors can be obtained in certain case with respect to the incremental solution. In this work, a Crystal Plasticity FEM model, based on dislocation densities, was used. As a first step, an aggregate of 20 grains of AISI 316L stainless steel under strain controlled cyclic loading was studied. Precise comparisons were conducted with incremental analysis and the results show that DCA seems to be an efficient solution in order to estimate the shakedown state of polycrystalline aggregates.

Particulate reinforced metal matrix composites (PRMMCs) are typical random heterogeneous materials whose global behavior depends on the microstructural characterisics. Recently a numerical approach was developed (Hachemi et al., Int J Plast 63:124–137, 2014 [1], Chen et al. Direct methods for limit and shakedown analysis of structures, 2015 [2]), by applying it to a typical PRMMC material WC/Co, we presented how the ultimate strength and endurance limit can be predicted from the material microstructures. Due to the randomness in the microstructures of PRMMCs, size of the representative volume element (RVE) has a nontrivial influence over the predicted effective behaviors. In order to understand how size of RVEs contribute to the result and based on that to eliminate its influence, a numerical investigation is performed in the present study. In this study, a large number of representative volume element (RVE) samples representing a representative PRMMC material, WC-20 Wt% Co, were built from artificial microstructures. The samples are obviously different in size, and by deploying the established numerical approach to these samples, ultimate strength and endurance limit were calculated. Afterwards, the derived material strengths were analyzed by multiple inferential statistical models. The statistical study reveals how strength and other effective material properties react to the change of the RVE size. On that basis, the study proposed a feasible and computationally inexpensive solution to minimize the size effect.

Direct methods aim to find the maximum load factor that a domain made of a plastic material can sustain before undergoing full collapse. Its analytical solution may be posed as a constrained maximisation problem, which is computationally solved by resorting to appropriate discretisation of the relevant fields such as the stress or velocity fields. The actual discrete solution is though strongly dependent on such discretisation, which is defined by a set of nodes, elements, and the type of interpolation. We here resort to an adaptive strategy that aims to perturb the positions of the nodes in order to improve the solution of the discrete maximisation problem. When the positions of the nodes are taken into account, the optimisation problem becomes highly non-linear. We approximate this problem as two staggered linear problems, one written in terms of the stress variable (lower bound problem) or velocity variables (upper bound problem), and another with respect to the nodal positions. In this manner, we show that for some simple problems, the computed load factor may be further improved while keeping a constant number of elements.

In this paper we propose a stochastic programming method to analyse limit and shakedown of structures under uncertainty condition of strength. Based on the duality theory, the shakedown load multiplier formulated by the kinematic theorem is proved actually to be the dual form of the shakedown load multiplier formulated by static theorem. In this investigation a dual chance constrained programming algorithm is developed to calculate simultaneously both the upper and lower bounds of the plastic collapse limit and the shakedown limit. The edge-based smoothed finite element method (ES-FEM) with three-node linear triangular elements is used for structural analysis.

The paper concerns mixed finite element models and experiments their capability in the analysis plastic collapse of both plane and three-dimensional problems respectively. The models are easy to formulate and implement because are based on simple assumptions for the unknown fields. A composite triangular or tetrahedral mesh is assumed over the domain. Within each element the displacement field is described by a quadratic interpolation, while the stress field is represented by a piece-wise constant description by introducing a subdivision of the element into proper sub-regions. The plastic collapse analysis is formulated as a mathematical programming problem and is accomplished by an Interior Point algorithm which furnishes both the collapse multiplier and the collapse mechanism. A series of numerical experiments shows that the proposed models perform well achieving the favorite context in plastic analysis, taking advantage of the absence of volumetric locking and the possibility of allowing discontinuities in the stress field within the element.

Shakedown theory has been recognised as a more rational basis for structural design of flexible road pavements. A lower-bound shakedown approach, which aims to find the maximum design load of a pavement structure, was developed by the University of Nottingham, that forms part of efforts among other researchers’ in applying shakedown theory in pavement designs. The lower-bound shakedown solutions were consistent with existing shakedown solutions assuming that the materials are isotropic and homogeneous following an associated plastic flow rule. Recently, this lower-bound approach was further developed to consider more realistic cases. Both two-dimensional and three-dimensional shakedown analyses were carried out taking into account cross-anisotropic or heterogeneous materials, the properties of which were programmed into a finite element software ABAQUS. For pavement materials obeying a non-associated flow rule, the corresponding two-dimensional lower-bound shakedown limits were also estimated by extending the lower-bound shakedown approach. A numerical step-by-step approach was also applied to address the non-associated problems and obtained similar results. Through these studies, influences of the original assumptions on the shakedown-based pavement designs can be assessed.

The present contribution aims at developing a numerical procedure for predicting the failure of high rise reinforced concrete walls subjected to fire loading conditions. The stability of such structures depends, on the one hand, on thermal strains inducing a curved deformed configuration and, on the other hand, on a local degradation of the constitutive material strength properties due to the increase of temperature across the wall thickness. A three step procedure is proposed, in which the yield design (limit analysis) method is applied on two separate levels. First, an up-scaling procedure on the wall unit cell is considered as a way for assessing the generalized strength properties of the curved wall, modelled as a shell, by taking into account reduced strength capacities of the constitutive materials. Secondly, the overall stability of the wall in its fire-induced deformed configuration is assessed using lower and upper bound based on shell finite elements and the previously determined temperature-dependent strength criterion. Second-order cone programming problems are then formulated and solved using state-of-the-art solvers. Different illustrative applications are presented to investigate the sensitivity of the wall stability to geometrical parameters. Finally, the influence of imperfect connections between panels is also considered using a simple joint behaviour.

To estimate the life of a structure, or a component, which are subjected to a cyclic loading history, the structural engineer must be able to provide safety margins. This is only possible by performing a shakedown analysis which belongs to the class of direct methods. Most of the existing numerical procedures addressing a shakedown analysis are based on the two theorems of plasticity and are formulated within the framework of mathematical programming. A different approach has recently appeared in the literature. It is rather more physical than mathematical as it exploits the physics of the asymptotic steady state cycle. It has been called RSDM-S and has its roots in a previously published procedure (RSDM) which assumes the decomposition of the residual stresses into Fourier series whose coefficients are found by iterations. RSDM-S is a descending sequence of loading factors which stops when only the constant term of the series remains. The method may be implemented in any existing FE code. It is used herein to establish shakedown boundaries for two-dimensional general loadings consisting of mechanical or thermomechanical loads.

Bree Interaction Diagrams have long been one of the major visual design guides for employing and evaluating shakedown in engineering applications. These diagrams provide representations of the realms in which elastoplastic behaviors, including shakedown, are found for a material and structure under variable loads. The creation of these diagrams often relies upon some combination of upper or lower bound shakedown theorems and numerical shakedown limit determination techniques. Part of the utility of these diagrams is that, for a given structure and loading conditions, inspecting them is sufficient to determine whether shakedown will occur or not. The diagrams cannot however, give the designer insight into how the conditions for shakedown are met. This chapter presents some graphical interpretations of one of the common methods for shakedown determination: the use of Melan’s Lower Bound Theorem. The intent is to provide additional insight for designers regarding how shakedown conditions are satisfied. In this way, additional directions for modifying designs to recover shakedown behavior may also be identified. Revisiting this well-established theorem from a graphical and pedagogical approach, also provides a foundation for interdisciplinary innovation. The particular focus is on simple examples that highlight ways in which Melan’s theorem may be applied to shakedown design problems.

In order to evaluate the safety and integrity of pressure vessels containing volume defects and piping with local wall-thinning at elevated temperature, a numerical method for plastic limit load of modified 9Cr-1Mo steel pressure vessel and piping is proposed in the present paper. The limit load of pressure vessel and piping at high temperature is defined as the load-carrying capacity after the structure has served for a certain time period. The power law creep behavior with Liu-Murakami damage model is implemented into the commercial software ABAQUS via CREEP for simulation, and the Ramberg-Osgood model is modified to consider the material deterioration effect of modified 9Cr-1Mo steel by introducing the creep damage factor into the elasto-plastic constitutive equation. For covering the wide ranges of defect ratios and service time periods, various 3-D numerical examples for the pressure vessels with different sizes of volume defects, the piping with local wall-thinning defects, and creep time are calculated and analyzed. The limit loads of the defected structures under high temperature are obtained through classic zero curvature criterion with the modified Ramberg-Osgood model, and the typical failure modes of these pressure vessels and piping are also discussed. The results show that the plastic limit load of pressure vessel and piping containing defect at elevated temperature depends not only on the size of defect, but also on the creep time, which is different from the traditional plastic limit analysis at room temperature without material deterioration.