Damage tensor

Fractures at different stress states have different growth model and a damage evolution law is also established for this case based on the arguments described above. Because the more growth of fractures indicates the more degradation of the rock masses, the damage tensor can be expressed as the following... 

In micromechanics (Muta 1982), tensor in Eq. 7.20 is defined as the eigenstrain, which is explicitly equivalent to the damage tensor in damage mechanics (Kachanov 1980). In seismology, the following tensor product is defined as a moment tensor,... 

Very few aromatic 77-radicals have been studied in the solid state. It has been stressed that magnetically dilute crystals are required, and these are not readily prepared. One very important example is that of a.a-diphenyl-jS-picrylhydrazyl. This was incorporated in small quantities in single crystals of the corresponding hydrazine and the 14N hyperfine and gf-tensors derived in the usual manner (Zeldes e.t al., 1960). This method of studying radicals, whilst normal for transition metal ions, is obviously inapplicable to most organic radicals whether stable or unstable. Fortunately, the method of radiation damage beautifully accomplishes this difficult task. This is discussed in Sections V and IX. 

Jones DK, Lythgoe D, Horsfield MA, Simmons A, Williams SC, Markus HS (1999) Characterization of white matter damage in ischemic leukoaraiosis with diffusion tensor MRI. Stroke 30 393-397... 

The damage of laminate is introduced by adding the damage contribution H tensor to the compliance tensor of composite (7, 8, 9). The only non zero component of H are H22 and //gg has no influence in the present analysis) ... 

The stress tensor for each point of the grain can be represented by one point in principal stress space. In that space, there exists a volume where the propellant is made worthless by significant damage, even possibly a crack. A major difficulty of such a representation is the fact that the failure properties of propellant depend strongly on loading conditions (temperature and strain rate). So in this paper, for each loading condition (one strain rate and one temperature), we construct a failure surface based on experimental data for several multiaxial stress states. 

Zhang H, Zhou C and Yi L. 2000. Damaging osmotic tensor of rock mass under hydraulic coupling conditions. Geotech. Engng. World, 3(3), pp. 11-14. 

The permeability tensor, being dependent upon the stress stale, changes with evolution of rock damage. The expressions for permeability tensor can be obtained for fractures subjected to different stress states and failure modes. 

The effective elastic stiffness tensor of damaged material C(d) is given by ... 

The fourth order tensor C is the derivative of the elastic tensor with respect to damage variable ... 

Following the considerations of Murakami and Ohno [104], damage is introduced as a second rank tensor. For a case of an open crack (transverse stress is tensile), energy equivalence of cracked and homogenous damaged media imposes the following reduction of the elastic constants ... 

Consequently the simulations produce a damage matrix for each apartment type model. Damage matrices relate each component s physical damage for a given wind speed and wind angle. A 4-dimensional tensor is therefore produced. See Eqn. (1). 

These results imply that the damage evolution and the accumulation of relative crack volume are also estimated from the moment tensor analysis. One example is given in Fig. 8.15. From moment tensors determined in bending tests of concrete specimens with a notch (Ohtsu and Ohtsuka 1998), the damage parameter and the accumulation of crack volume are estimated. In the case of the center-notched specimen (type CC), the damage suddenly increases, while the damage evolution is gradual in the off-center notched specimen (type OC). 

Fig. 8.15. Damage evolution analyzed from the moment tensor analysis.

Ohtsu M, Okamoto T, Yuyama S (1998) Moment tensor analysis of acoustic emission for cracking mechanisms in concrete. ACI Stracmral Journal 95(2) 87-95 Ohtsu M, Ohtsuka M (1998) Damage evolution by AE in the fracture process zone of concrete. J Materials, Cone. Struct. Pavement, JSCE, 599/V-40 177-184 Ohtsu M (2000) Moment tensor analysis of AE and SiGMA code. Acoustic Emis-sion-Beyond the Millennium. Elsevier, pp 19-34 Ouyang C, Landis E, Shah SP (1992) Damage assessment in concrete using acoustic emission. Nondestructive Testing of Concrete Elements and Structures, ASCE, pp 13-24... 

Shiotani T, Nakanishi Y, Iwaki K, Luo X, Haya H (2006) Evaluation of reinforcement in damaged railway concrete piers by means of acoustic emission. AEWG, Journal of Acoustic Emission 23 260-271 Yuyama S, Okamoto T, Shigeishi M, Ohtsu M (1994) Quantitative evaluation and visualization of cracking process in reinforced concrete by a moment tensor analysis of acoustic emission. ASNT, Materials Evaluation 53 751-756... 

Diffusion tensor imaging (DTI) and quantitative T-2 MRI were used to characterize ex vivo the white matter damage at 3 and 8 weeks following dorsal column transection (DC Tx) injury at the cervical level C5 of rat spinal cords. " This study demonstrates that quantitative MRI can accurately characterize white matter damage in DC Tx model of injury in rat spinal cord. 

Other approaches have been suggested to more formally define the form of damage that occurs during fatigue (34-37). These methods model damage as an array of microcracks (34,35) that inevitably require a tensoral definition of... 

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