Issue 75

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

(a) (c) Figure 2: Identification of elastic, elasto-plastic, and plastic limits for a representative truss: (a) loading and support scheme; (b) load– plastic deformation response; (c) load–displacement response. Geometrical nonlinear finite element analysis Building upon the previously established linear-elastic and elasto-plastic framework, this section extends the analysis to account for geometric nonlinearities that arise in structures undergoing large deformations. In certain structural systems, such as slender truss configurations, large displacements and rotations may significantly affect the mechanical response. To accurately capture such behavior, the analysis must incorporate geometric nonlinearity within a total Lagrangian finite element framework. In this formulation, the Green–Lagrange strain tensor ij  and the second Piola–Kirchhoff stress tensor ij s are employed. These quantities form a work-conjugate pair suitable for the analysis of large deformations while remaining (b)

consistent with the underlying constitutive laws. The Green–Lagrange strain tensor is defined as:   , , , , ij i j j i k i k j u u u u    

1 2

(18)

where u denotes the displacement field, and indices i , j , k refer to spatial directions. In finite element implementations, the incremental form of the strain–displacement relationship is written as:   d d  ε B U U (19) where   B U is the displacement-dependent strain–displacement matrix and d U is the vector of incremental nodal displacements. The matrix   B U evolves with the current configuration, capturing the geometric nonlinearity of the deformation process. In consistency with the previously presented Eqn. (2), the constitutive relation in terms of the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor is expressed as:

ijkl kl C  

s

(20)

ij

To assess equilibrium in the deformed configuration, the global residual vector is defined as:

  0 T dV   R U P B s

(21)

where 0 P is the vector of externally applied loads, and s represents the current second Piola–Kirchhoff stress field. The residual vector   R U quantifies the imbalance between internal and external forces and vanishes when equilibrium is satisfied. The nonlinear equilibrium equations are solved iteratively using the Newton–Raphson method, in which the tangent stiffness matrix is computed as:

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