A dislocation-based finite element method for plastic collapse assessment in solid mechanics

In most structural analyses, the limit load and the required ultimate displacement before the loss of functionality are the desired results from the structural calculation. Accurate time-history analysis of structures is possible only when the load path to which the structure is subjected is known in detail. In the majority of practical applications, on the contrary, the load path is unknown, and only the load intensity range is fixed by statistical evaluation of the bounds of the load intensity, see, for example, the American Society of Civil Engineers documentation [1]. The limit analysis allows calculating the ultimate load factor of the structures from the knowledge of the load pattern and the structural geometric and constitutive properties, even if the load time history is not known. Indeed, the load safety threshold calculation is based on the knowledge of the domain of the loads, whose dimensions are defined by the load multiplier, i.e., the locus to which the loads must belong to satisfy the material strength requirements. To apply the limit design to the structural calculation through the finite element method (FEM), it is crucial to define the way one can represent the set of the eigenstress in the structures as a linear span of discrete parameters. The eigenstress set is the null space of the equilibrium operator of the structure and is related to eigenstrain, which represents any imposed deformation depending on anelasticity [2].

The application of limit analysis and plasticity to structural safety concerns a wide range of engineering fields [3]. It was the main topic of Prager, Drucker, and Greenberg’s works [4‐8]. Massonnet and Save [9] address the plastic response of structures introducing the collapse calculation for one-dimensional beam assembly and constitute the starting point of several researches [10].

A twofold approach to the limit analysis is found in the literature. The first, the kinematic method, consists in finding the collapse load as the lower bound among the loads in equilibrium with the stress according to a compatible collapse mechanism. The second, the static approach consists in finding the upper bound among the loads in equilibrium with one of the admissible stresses (defined as the combination of one of the eigenstress stresses and the stress in equilibrium with the loads under the elastic hypothesis).

In [11, 12], the plasticity of structures has been widely studied in primis with reference to monotonic load paths. In [13‐16], the load path is assumed to be randomly variable, so the shakedown is addressed using mathematical programming algorithms. In soil mechanics, the limit analysis through the plasticity methods is widely applied; see Chen [17]. In the works [18, 19], the lower-bound and upper-bound theorems for geo-mechanics are formulated.

In the fields of porous media and bio-mechanics, the limit equilibrium is used as a primary tool to assess the fracture risk of prosthesis implants and corresponding optimization strategies as reported in [20‐23].

The numerical formulation of the plasticity method can be found in the books [24‐26], where iterative step-by-step analyses through the finite elements are discussed in detail. Strain concentration and large displacement in pipe limit analysis addressed via variational approach can be found in [27]. The book [28] is entirely devoted to the inelastic behavior of structures, where the algorithms of computational plasticity are described. In [29‐32], an iterative method is proposed to calculate the shakedown of elastic-perfectly-plastic structures and a fast incremental-iterative solution method suitable for the FEM analyses of large structures. Furthermore, [33], using the boundary element method, evaluates the eigenstress stress through an iterative algorithm and integral equations.

A semi-analytical approach is proposed in [34, 35] where the eigenstress stress for axial symmetrical domes is evaluated, and a comparison with the experimental campaign and crack lines analytical calculations reported in [36, 37] is discussed highlighting the good confidence of the method with experiments. There are several procedures for calculating the limit load of structures in terms of both one-dimensional and two-dimensional limit analysis based on the assumption that the material constituting the structures has no resistance to tension. The applicability of the limit analysis to this family of materials, called no-tension materials, i.e., concrete, stone, soils, and ceramics, can be found in [38], where a complete theoretical framework is provided. Many pioneering works deal with the limit analysis of no-tension materials [39, 40], which has been subsequently implemented through the finite elements, [41‐44] and [45]. In [46, 47], a critical evaluation of state-of-the-art computational methods concerning variational formulations, selection of interpolation spaces, numerical solution strategies, and numerical stability. Although these arguments are interconnected, a topic-oriented presentation is employed to thoroughly examine the abundant available literature. Lastly, the performance of several significant mixed finite element formulations is investigated in numerical simulations. The automated computation of limit analysis problem is described in [48] The optimization problem is solved numerically using the interior-point solver Mosek, taking advantage of the conic representation for yield criteria. The paper demonstrates the effectiveness of the approach through illustrative examples, spanning classical continuum limit analysis problems to generalized mechanical models. . A class of strain–displacement elements in upper-bound limit analysis is presented in [49]. The paper stated that only linear and quadratic displacement elements with straight sides, generating constant and linear strains, meet specific requirements. The text theoretically and practically demonstrates the existence of a broader class of triangular elements with straight sides, based on Bernstein polynomials, capable of providing rigorous upper bounds. This class allows for polynomial orders without upper restrictions and accommodates both continuous and discontinuous displacements. In [50], a non-incremental energy-based algorithm is proposed to calculate the equivalent material’s distribution and orientation, to achieve a compression-only state of stress. A finite element procedure where the structure is discretized using rigid infinitely resistant eight-node hexahedral elements and nonlinear interfaces, exhibiting deterioration of the mechanical properties, is proposed in [51].

The aim of the present work is to propose a general procedure for formulating the limit design through the FEM. The proposed formulation assumes that the strain parameters govern the phenomenon. The method is based on the calculation of the limit load for collapse or shakedown through the permanent strain. To this scope, Volterra’s dislocation, from which the name of the procedure, VFEM, at the element level has been assumed to represent the permanent strain, which has been described numerically by displacement-like coordinates. The whole procedure results in the definition of a numerical linear operator named the dislocation matrix, which maps the nodal dislocation parameters and the constraint displacements on the eigenstress stress. The proposed representation does not depend on the constitutive equation of the material. Furthermore, it can be seen that in the analyzed cases the principal values of the dislocation matrix are related to the multiplicity of the equilibrium equation solutions. The proposed representation is useful to generalize the construction of the eigenstress stress space in the FEM and the implementation of the lower bound theorem of the limit analysis, both for monotonic and random variable loads within prescribed limits [26, 52]. In the following, the finite element formulation of the dislocation-based limit analysis is described the eigenstress stress representation in terms of dislocations and constraint is derived, and the constrained optimization problem is described. The procedure has been evaluated through comparison with standard examples from the literature. The proposed procedure implements the direct application of the lower bound theorem of the limit analysis in the Mélan form. It allows the calculation of the collapse and the shakedown limits of a structure under several load patterns through pure elastic calculation. The key aspect of the proposed method is to generalize the static theorem application as used for framed structures and trusses to any discretization through finite elements of articulated 2D and 3D structures. Moreover, the procedure is based on elastic calculations only and does not require step by step, iterative, or predictor-corrector strategies.

In the following, 2D structures have been considered. Namely, a Holed Plate with tensile loads, a cantilever beam with different geometry, a clamped simply-supported beam with a transversal load, and a pipe annular cross-section subjected to internal pressure. The method has also been applied to porous media modeled as a clamped plate with several random holes. Finally, the numerical evaluation of the strength of a porous elementary volume has been addressed.

This section presents a procedure to evaluate the collapse and the shakedown load of structures using FEM representation of eigenstress, i.e., the residual stress due to dislocation arising. The procedure uses a finite element description of Volterra’s dislocation. The approach is devoted to linking the permanent strain that arises from the ultra-elastic behavior of the structure to the nodal parameters governing the growth of dislocations. In particular, it is shown hereafter that the permanent strain and the corresponding eigenstress stress can be represented in terms of pseudo-displacement and constraint nodal parameters. Calculating the eigenstress stress allows the formulation of the limit load assessment algorithm in terms of eigenstress according to Melan’s theorem.

In the present section, some examples have been analyzed considering the 2D model of plates. The structures have been discretized using Lagrange’s elements with four nodes and linear interpolation of the displacement. The proposed examples presented no evidence of convergence problems due to the absence of slenderness effects, nonlinear geometric effects, and constitutive nonlinearity. The constraints of the optimization program have been written in terms of the corresponding stress components. Equation (16) contains only elastic quantities calculated at nominal load intensity. Neither elastic–plastic results are required nor iterative predictor–corrector strategies. The whole process reduces to a constrained maximum problem that is solved using mathematical programming algorithms. In the examples, ©MATLAB optimization tools have been used, and in particular linear programming has been implemented through the linearization of the inequalities described above. The linearization allowed us to simplify the solution and apply simple convergent algorithms.

The work has presented the direct application of the lower bound theorem of limit analysis via finite elements employing Mèlan’s formulations for determining the collapse load of structures subjected to either monotonic or randomly variable loads. A methodological procedure is expounded that hinges upon the multiplicity of solutions to the equilibrium equations under the specified loading conditions. A comprehensive \(\varvec{\mathcal {V}}\) FEM procedure is outlined, delineating the representation of equilibrated stress’s linear variety as a composite of the linear variety spanned through \(\varvec{\mathcal {V}}\) and a particular solution postulating the structure to possess indefinite elasticity

The limit analysis of the structure is executed through Mèlan’s theorem, leveraging nodal parameters of dislocations and constrained displacements as optimization design variables. The formulation is characterized by its generality, accommodating linear-elastic perfectly-plastic materials with isotropic, anisotropic, iso-resistant, and uniso-resistant constitutive laws. The establishment of the limit domain of admissible stresses relies on the Mises–Scheicher equation, followed by the linearization of the limit locus to enhance accuracy, coupled with the introduction of a penalty coefficient for actual stress.

Validation of the method is performed across canonical geometries and constitutive materials, further extending its application to analyze plates with various hole geometries and multi-hole plates. Emphasis is placed on the necessity of estimating dissipated energy during loading and the method’s limitations concerning the control of ductility requirements. To address these concerns, a proposal is made to utilize \(\varvec{\mathcal {V}}\)FEM for upper-bound calculations of dissipated energy and local permanent displacements, showcasing the potential for future applications and developments in the field.

The works [60, 61] show that the proposed eigenstress map, in terms of displacement and dislocation, has its natural evolution toward the application to the assessment of upper bounds of the dissipated energy and the local permanent displacements.