Damage-tensor-based nondimensional parameters governing secondary faulting behavior

2013 ◽  
Vol 600 ◽  
pp. 205-216 ◽  
Author(s):  
Takehito Suzuki
Keyword(s):  
Damage Tensor

Damage tensor theory and its application to tunnelling

2020 ◽  
pp. 51-58
Author(s):  
G. Swoboda ◽  
M. Stumvoll ◽  
Han Beichuan
Keyword(s):  
Tensor Theory ◽  
Damage Tensor

scholarly journals Damage Tensors and Tensors of Continuity in Stress Analysis

1983 ◽  
Vol 38 (12) ◽  
pp. 1383-1390 ◽  
Author(s):  
J. Betten
Keyword(s):  
Linear Operator ◽  
Coordinate System ◽  
Stress Tensor ◽  
Constitutive Equations ◽  
Symmetric Tensor ◽  
Diagonal Form ◽  
Symmetric Part ◽  
Third Order ◽  
Symmetric Property ◽  
Damage Tensor

Abstract Starting from a third order skew-symmetric tensor of continuity to represent area vectors (bivectors) of Cauchy's tetrahedron in a damaged state, a second order damage tensor is found which has the diagonal form with respect to the considered coordinate system. The second part of the paper is concerned with the stresses in a damaged continuum. Introducing a linear operator of rank four a net-stress tensor is formulated. This tensor can be decomposed into a symmetric part and into an antisymmetric one, where only the symmetric part is equal to the net-stress tensor introduced by Rabotnov [7].In view of the formulation of constitutive equations the non-symmetric property of the actual net-stress tensor is a disadvantage. Therefore, a pseudo-net-stress tensor is introduced, which is symmetric.

Finite Element Modelling of Volumetric and Shear Ductile Micro- and Macro-Fracture Processes Under Long Time Loading

2008 ◽  
Author(s):  
Lucija Pajic ◽  
Alexander A. Lukyanov
Keyword(s):  
Finite Element ◽  
Pipeline Steel ◽  
Damage Model ◽  
Shear Loading ◽  
Continuous Operation ◽  
Propagation Path ◽  
Economic Implications ◽  
Long Time ◽  
Fracture Processes ◽  
Damage Tensor

Submarine and onshore pipelines transport enormous quantities of oil and gas vital to the economies of virtually all nations. Any failure to ensure safe and continuous operation of these pipelines can have serious economic implications, damage the environment and cause fatalities. A prerequisite to safe pipeline operation is to ensure their structural integrity to a high level of reliability throughout their operational lives. This integrity may be threatened by volumetric and shear ductile micro- and macro-fracture processes under long time loading or continuous operation. In this paper a mathematically consistent damage model for predicting the damage in pipeline structures under tensile and shear loading is considered. A detailed study of widely used damage models (e.g., Lemaitre’s and Gurson’s models) has been published in the literature. It has been shown that Gurson’s damage model is not able to adequately predict fracture propagation path under shear loading, whereas Lemaitre’s damage model (Lemaitre, 1985) shows good results in this case (e.g., Hambli 2001, Mkaddem et al. 2004). The opposite effect can be observed for some materials by using Gurson’s damage model in the case of tensile loading (e.g., Tvergaard and Needleman 1984; Zhang et al. 2000; Chen and Lambert 2003; Mashayekhi et al. 2007) and wiping die bending process (Mkaddem et al. 2004). Therefore, the mathematically consistent damage model which takes into account the advantages of both Lemaitre’s and Gurson’s models has been developed. The model is based on the assumption that the damage state of materials can be described by a damage tensor ωij. This allows for definition of two scalars that are ω = ωkk/3 (the volume damage) (Lukyanov, 2004) and α = ωij′ωij′ (a norm of the damage tensor deviator ωij′ = ωij −ωδij) (Lukyanov, 2004). The ω parameter describes the accumulation of micro-pore type damage (which may disappear under compression) and the parameter α describes the shear damage. The proposed damage model has been implemented into the finite element code ABAQUS by specifying the user material routine (UMAT). Based on experimental research which has been published by Lemaitre (1985), the proposed isotropic elastoplastic damage model is validated. The results for X-70 pipeline steel are also presented, discussed and future studies are outlined.

Damage Deactivation

1995 ◽  
Vol 62 (2) ◽  
pp. 450-458 ◽  
Author(s):  
N. R. Hansen ◽  
H. L. Schreyer
Keyword(s):  
Damage Mechanics ◽  
Continuum Damage Mechanics ◽  
Uniaxial Stress ◽  
Projection Operators ◽  
Continuum Damage ◽  
Order Tensor ◽  
Smooth Functions ◽  
One Dimensional ◽  
Opening And Closing ◽  
Damage Tensor

A phenomenological algorithm, motivated by the “mode I” microcrack opening and closing mechanism, is developed for the deactivation and reactivation of the damage effects as modeled by certain continuum damage mechanics theories. One-dimensional formulations with and without coupled plasticity are used to elucidate concepts associated with damage deactivation and to suggest multidimensional deactivation formulations applicable to continuum damage theories that employ a second-order tensor as the damage measure. Strain-based projection operators are used to deactivate the damage effects in the damage tensor. Motivated by observations from one-dimensional coupled formulations, both the total and elastic strains must be compressive for the damage to be rendered inactive. By introducing smooth functions to represent the transition from the active to the inactive state, discontinuities in the response are avoided. To illustrate the aspects associated with deactivation, a consistent set of examples using a strain-controlled one-cycle uniaxial stress loading is given for each formulation.

Kinematic Description of Damage

1998 ◽  
Vol 65 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Taehyo Park ◽  
G. Z. Voyiadjis
Keyword(s):  
Fourth Order ◽  
Damage Mechanics ◽  
Effective Strain ◽  
Continuum Damage Mechanics ◽  
Second Order ◽  
Finite Strains ◽  
Damage Effect ◽  
Energy Equivalence ◽  
Simple Mapping ◽  
Damage Tensor

In this paper the kinematics of damage for finite elastic deformations is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. However, the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses. One uses either the hypothesis of strain equivalence or the hypothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a relation between the effective strain and the damage elastic strain that is also applicable to finite strains. This is accomplished in this work by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. The proposed approach shows that it is equivalent to the hypothesis of energy equivalence for finite strains. In this work, the damage is described kinematically in the elastic domain using the fourth-order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. The constitutive equations of the elastic-damage behavior are derived through the kinematics of damage using the simple mapping instead of the other two hypotheses.

scholarly journals Geometric Damage Tensor Based on Microplane Model

1991 ◽  
Vol 117 (10) ◽  
pp. 2429-2448 ◽  
Author(s):  
Ignacio Carol ◽  
Zdeněk P. Bažant ◽  
Pere C. Prat
Keyword(s):  
Microplane Model ◽  
Damage Tensor
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