Issue 75

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

1     n n i i i i n max i Al A l    

   

   

A l

 

min i



1





f G

(24b)

s

n

A l

min i



i

i

1

1

 P P

, if m P

0, m

   

0

  P

1 p m

(24c)

 P P



if m

1

 

0

P

0



if    

if

0,

stab

  

  2 

p

(24d)



 



1

,

 

stab



stab

if W W 

0,

0    p   

p

, p max

  p

p p W W W  

(24e)

, p max

3

if W W 

,

p

, p max

W W

, p max

Subject to:

U U 

(24f)

max k

max

k

n

l

1

  2 R i



i



N W 

W

(24g)

p

p

0

1 E A  i

2

i

M

   x

0   X X x where

 

(24h)

j

j

j

  3 p p W is introduced to constrain the extent of plastic deformations.

As shown in Eqn. (24a), a new penalty term

p W exceeds

According to Eqn. (24e), this penalty becomes active when the complementary strain energy of residual forces

, p max W . In such cases, the penalty is calculated using a normalized expression that depends on the

the predefined threshold

attained value of 0 p W . This normalization ensures that   3 p p W remains bounded within the interval   0,1 , thereby maintaining consistent scaling across all penalty components. The inclusion of this term allows for precise control over the admissible inelastic behavior while ensuring compatibility with the overall design objectives and structural safety criteria. In addition to the plasticity-related term   3 p p W , the penalty function   1 p m P continues to play an essential role in the elasto-plastic design formulation, as structural failure may still occur without the development of plastic deformations— most notably due to elastic instability, such as buckling. However, as shown in Eqn. (24c), the formulation of   1 p m P differs from its counterpart in Eqn. (23c), which was originally defined for the purely elastic design case. In the present elasto-plastic context,   1 p m P is activated only if the structure fails to reach the predefined load level 0 P during the loading history. This modification reflects the relaxation of the strict elastic requirement, as load levels exceeding el P are now permitted. Consequently, the objective shifts toward ensuring that the target load 0 P is achieved, while the extent of inelastic deformation is explicitly controlled through the complementary strain energy p W , evaluated from the residual internal forces. p W under the applied load level m P , the allowable , p max W , and

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