PSI - Issue 42

T. Fekete et al. / Procedia Structural Integrity 42 (2022) 1684–1691

1687

4

T. Fekete et al.: Extending reliability of FEM simulations… / Structural Integrity Procedia 00 (2019) 000–000

T = C F F , while the Green-Lagrangian strain tensor is

The right Cauchy-Green tensor, C is specified by

(

)

1 2 = − E C I –see Bažant and Cedolin (1991), Hirschberger (2008) and Clayton (2011)–.

determined as

X X = ∇ = ∇ H u u  ɺ –the displacement

When the displacement vector u is regarded as primary variable, then

= − H F I . In this approach, the Green-Lagrangian

gradient– enters the picture, which is connected to F via

deformation tensor has the following formulation –see e.g., Hirschberger (2008), Papenfuß (2020)–:

(

)

T = + + ⋅ = + E H H H H E E ɶ ɶ T lin

(4)

1 2

nonlin

(

)

(

)

E ɶ

E ɶ

1 2 = ⋅

+

1 2 = ⋅

⋅

with

and

. Relation (4) presents a Boltzmann continuum model for

lin

T

nonlin

T

H H

H H

describing geometrically nonlinear, large deformations. When only the lin E ɶ part of the Green-Lagrange tensor is used, one is talking about linearized theory of deformations. From physics point of view, the use of large deformation theory paves the way for the incorporation of kinematic nonlinearities into computations, which, together with the appropriate material nonlinearities, allows a more accurate description of material behavior in situations where local strains are well-known to be large, e.g., at final rupture of a specimen, –see Clayton (2011)–. The mechanical state of the body and its time evolution can be described, when the kinematic model is complemented by a system of field equations, describing the interactions of the body and their dynamics. The field equations form a specific system, consisting of a set of general balances, valid for all material models, complemented by a set of constitutive equations describing the behavior of a given material –see e.g., Béda, Kozák and Verhás (1995), Muschik, Papenfuß and Ehrentraut (2001) and Papenfuß (2020)–. Using the kinematical model presented in Eq. (4), a complete system of equations governing the behavior of the system has the following structure: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 , , kinematicequation , , , strain decomposition theorem , , , balance of momentum , , balance of moment of momentum , , : , balan X X X X E p X T d d d d W τ τ τ τ τ τ ρ τ τ τ τ τ τ τ τ = ∇ +∇ + ∇ ⋅ ∇ = + +∇ ⋅ = = = ε X u u u u X ε X ε X ε X v X σ X f X σ X σ X X σ X ε X ⌢     ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ɺ ɺ ce of mechanical power (5) In these equations, ρ denotes mass density, ε ⌢ stands for the Green-Lagrange deformation tensor –expressed in terms of the displacement vector u –, σ ⌢ is the true stress tensor, d τ v denotes acceleration, f means density of external forces, W denotes the stored mechanical energy, W ɺ stands for the mechanical power, ∙ means inner multiplication of two vectors, : represents the inner multiplication of two tensors, ( ) E E d = ε F σ stands for the elastic and ( ) p p d = ε F σ for the plastic constitutive equation. Note that these constitutive equations do not contain an explicit time dependence, which means that the material response to a load is assumed to be immediate. As mentioned above, in engineering practice, strength calculation methods based on linearized deformation theory are generally used, because this approach was approved in standards decades ago, –see e.g., the ASME Code, the RCCM Code, the DIN or the ISO instructions–. Note that this simplified approach reflects the understanding of CMS towards the end of the 19 th century –Maugin (2009)–. As the practical experience of the past century has largely shown, components and equipment designed –i.e., dimensioned and verified by the DSC s– using such methods usually have varying degrees of residual strength and/or expected lifetime reserves. However, in recent decades, engineering practice has also raised several issues requiring the tracking of large inelastic deformations, including: • Simulations of manufacturing processes involving large plastic deformations (e.g., forging); • Accidental situations where deformations and stresses are well above the design/operational limits and simulations aim at establishing the safety limits for human life and the environment (e.g., car accidents); • Elastic-plastic fracture mechanics simulations, where the global strains and stresses may remain small, but the strains and stresses along the crack front are large (e.g., SIC s of a large-scale equipment). ( ) E ⌢ ( ) p ⌢ , constitutive relations E p d d = ε F σ = ε F σ ⌢ ⌢