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Generalization of T and A integrals to time-dependent materials: analytical formulations

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Abstract

This paper deals with the generalization of T-integral to crack growth process in viscoelastic materials. In order to implement this expression in a finite element software, a modelling form of this integral, called Aθ, is developed. The analytical formulation is based on conservative law, independent path integral, and a combination of real, virtual displacement fields, and real, virtual thermal fields introducing, in the same time, a bilinear form of free energy density F. According to the generalization of Noether’s method, the application of Gauss Ostrogradski’s theorem combined with curvilinear cracked contour, T v is obtained. By introducing a volume domain around crack tip, the modelling expression Aθ is also defined.. Finally, the viscoelastic generalization through a thermodynamic approach, called A v , is introduced by using a discretisation of the creep tensor according to a generalized Kelvin Voigt representation.

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Abbreviations

a, Δa:

Crack length and crack increment

A1A2, B1B2:

Crack lips

A(Γ1), A(Γ2):

Areas enclosed by Γ1 and Γ2

A :

A-integral

Aθ :

Modeling form of A-integral

Aθ v :

Viscoelastic form of Aθ-integral

A v :

Generalization of A-integral to viscoelastic behavior

b :

Width specimen

F :

Helmholtz strain energy density

F • :

Bilinear form of strain energy density

G v , G e :

Viscoelastic and elastic energy

G W , G s :

External and fracture energy

I1, I2:

Invariant notations

J ijkl :

Four order creep compliance tensor

J :

Rice’s integral

K :

Kinetic energy

\({^{u}K_{I}^{(p)}, ^{v}K_{II}^{(p)}}\) :

Real and virtual stress intensity factors in opening mode and shear mode induced by p th spring

\({k_{ijkl}^p}\) :

Spring rigidity components

\({\vec{n}}\) :

Normal vector of components n j

\({\vec{n}_1, \vec{n}_2}\) :

Normal vectors to contours Γ1 and Γ2

N :

Total number of Kelvin Voigt cells

\({s_j^{(p)}}\) :

Roots of characteristic equation

\({S_{11}^{(p)}, S_{22}^{(p)}, S_{12}^{(p)}, S_{33}^{(p)}}\) :

Compliance tensor components

S :

Surface domain

\({t, \xi, \phi, \varsigma}\) :

Time variables

t c :

Critical time

T :

T-integral

T p :

New analytical form of T-integral

T v :

Generalization of T-integral to viscoelastic behavior

u, v:

Real and virtual displacement fields of components u i and v i

u(p), v(p):

Real and virtual displacement fields in the p th spring

U e :

Elastic strain energy Voigt cell

W vis :

Viscous energy

W S :

Fracture energy

V :

Volume domain

p :

Spring of pth Kelvin

x1, x2:

Axis

α (x1, x2, t):

Spatial and temporal function

α t :

Expansion thermal coefficient

δL :

Lagrangian variation

\({\delta\tilde{u}, \delta\tilde{v}}\) :

Real and virtual Euleurian representation

\({\delta{u}^{{\ast}}, \delta{v}^{{\ast}}}\) :

Real and virtual Lagrangian representation

\({\eta_{ijkl}^p}\) :

Dash-pot viscosities components

λ, μ :

Lame’s coefficients

\({\vec{\theta}}\) :

Vector field

σ ij , ε ij :

Elastic stress and strain tensor component

\({\sigma_{ij}^{u}, \sigma_{ij}^v}\) :

Total real and virtual stress components

\({^{(p)}\sigma_{ij}^{u}, ^{(p)}\sigma_{ij}^{v}}\) :

Real and virtual stress components in the p th spring

∂V :

Contour around the crack tip

Δt :

Time increment

Γ1, Γ2 :

Surface contours

Ω:

Integration domain

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Authors and Affiliations

  1. Laboratoire de Mécanique et Ingénierie, EA 3867, Clermont Université, Université Blaise Pascal, BP 10448, 63000, Clermont-Ferrand, France

    Rostand Moutou Pitti

  2. GEMH Laboratory, Limoges University, Centre Universitaire Génie Civil, Boulevard Jacques Derche, 19300, Egletons, France

    Frédéric Dubois & Christophe Petit

Authors
  1. Rostand Moutou Pitti
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  2. Frédéric Dubois
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  3. Christophe Petit
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Correspondence to Rostand Moutou Pitti.

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Moutou Pitti, R., Dubois, F. & Petit, C. Generalization of T and A integrals to time-dependent materials: analytical formulations. Int J Fract 161, 187–198 (2010). https://doi.org/10.1007/s10704-010-9453-1

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  • DOI: https://doi.org/10.1007/s10704-010-9453-1

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