Issue 75

P. Grubits et alii, Fracture and Structural Integrity, 75 (2026) 124-156; DOI: 10.3221/IGF-ESIS.75.10

within an acceptable limit under the applied load. In addition, a supplementary optimization case is conducted in the elasto plastic scenario by imposing displacement thresholds to assess the influence of serviceability requirements. In the 25-bar truss example, the analysis focuses exclusively on elasto-plastic design, where two cases are compared: one without displacement constraints and another where serviceability limitations are explicitly enforced. The obtained results confirm the efficiency of the developed framework, underlining the practical relevance of advanced optimization techniques in improving structural design precision and achieving safer, more materially efficient configurations.

T HEORETICAL FOUNDATION FOR MATERIAL AND GEOMETRICAL NONLINEAR ANALYSIS

T

his section outlines the fundamental theoretical components underlying the proposed framework. First, the complementary plastic work theory is introduced, which serves as the methodological basis by providing a physically meaningful measure of plastic deformations. Building on this foundation, the principles of elastic and elasto-plastic limit analysis are discussed, with particular emphasis on the role of complementary strain energy associated with residual forces. Finally, the core aspects of geometrically nonlinear analysis are presented, as incorporated into the structural design process to accurately reflect the behavior of structures under large deformations. Complementary plastic work theory This section outlines the theoretical basis of the complementary plastic work theory. Consider a model composed of an elasto-plastic material, assumed to be independent of both time and temperature. The body occupies a volume V and is bounded by a surface S , which is partitioned into two disjoint subsets: u S , where zero surface displacement is prescribed, and q S , which is subjected to quasi-static surface tractions   , i q t x , with i x denoting the position vector on the surface. Following the foundational work of Kaliszky [6], the field quantities at a given time t are defined as follows:   ij t   actual stress components,   ij t  and   i u t = actual strain and displacement fields,   el ij t   fictitious stress components corresponding to an ideal purely elastic response,   el ij t  and   el i u t  fictitious elastic strains and displacements associated with   el ij t  . To further characterize the internal force state, two additional stress fields are introduced:   R ij t   the actual residual stress distribution, R ij   any arbitrary, time-independent self-stress distribution. According to the constitutive relation for time-independent elasto-plastic materials, the total strain ij  can be additively decomposed into elastic and plastic components:

ε ε ε el pl ij ij ij  

(1)

This relation is assumed to hold at every time step throughout the loading history. Consequently, the material behavior does not involve any intrinsic time dependence, and the evolution of inelastic strains can be fully described within a rate independent framework. The elastic strain is related to the actual stress tensor via the fourth-order elastic stiffness tensor ijkl C , assuming linear elasticity:

el

ij ijkl kl C   

(2)

Meanwhile, the evolution of plastic strain is governed by the associated flow rule [6]:



f









pl



0   if f





,  



and f

otherwise

ε 

0

0,

0

(3)

ij



ij 

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