Search results

Numerical solutions in computational plasticity are severely challenged when concrete and geomaterials are considered with non‐regular yield surfaces, strain‐softening and non‐associated flow. There are two aspects that are of immediate concern within load steps which are truly finite: first, the iterative corrector must assure that the equilibrium stress state and the plastic process variables do satisfy multiple yield conditions with corners, Fi(σ, q) = 0, at discrete stages of the solution process. To this end, a reliable return mapping algorithm is required which minimizes the error of the plastic return step. Second, the solution of non‐linear equations of motion on the global structural level must account for limit points and premature bifurcation of the equilibrium path. The current paper is mainly concerned with the implicit integration of elasto‐plastic hardening/softening relations considering non‐associated flow and the presence of composite yield conditions with corners.

A simple beam element is developed for the solution of large deflection problems. The total Lagrangian formulation is based on the kinematic relations proposed by Reissner for finite rotations and stretching as well as shearing of plane beams. The motion is discretized by linear expansions of the global displacement components and the cross‐sectional rotation in two‐dimensional Euclidean space yielding a simple beam element with three degrees of freedom at the two nodes. The shear locking is reduced by selective integration in order to eliminate the spurious shear constraint similar to interdependent variable interpolation. The large rotation formulation is compared with two forms of moderate rotation theories which have been used in the past to develop the geometric stiffness properties for linear stability analysis of the so‐called Mindlin plate elements. The predictive value of different geometric stiffness approximations is assessed with several examples which range from the static and kinetic stability analysis of the classical Euler‐column to the large deflection problem of a clamped beam.

In this paper, we examine the influence of the third invariant in computational plasticity. For this purpose we consider the extended Leon model, an elasto‐plastic model for concrete materials which accounts for the difference of shear strength in triaxial compression and triaxial extension. Consequently, the deviatoric trace of the loading surface is no longer circular like in von Mises and Drucker‐Prager plasticity. In the limit it approaches the triangular shape of the Rankine condition of maximum direct stress. Thereby, elliptic functions describe the out‐of‐roundness of the circular trace in terms of C¹‐continuous functions of the Lode angle. The algorithmic aspects of the third invariant considerably complicate the computational implementation since the radial return method of J₂‐plasticity does no longer maintain normality leading to loss of deviatoric associativity. The paper will focus on the computational issues near the three regions with high curvature at the compressive meridians with special attention on the lack of convergence of the plastic return algorithm and its slow rate of convergence in these regions. The algorithmic discussion at the constitutive level will be augmented by the axial plane‐strain compression test in order to illustrate the effect of the third invariant at the structural level of finite element analysis.

In this study we examine the spectral properties of stiffness degradation at the constitutive level and at the levels of finite elements and their assemblies. The principal objective is to assess the effects of defects on the elastic stiffness properties at different levels of observation. In particular, we are interested in quantitative damage measures, which characterize the fundamental mode of degradation in the form of elastic damage at the level of constitutive relations and at the level of finite elements and structures.