Stochastic Internal Stresses in Heterogeneous Materials

Kreher, W.; Pompe, W.

doi:10.4028/www.scientific.net/msf.62-64.747

Cited by 46 publications

(94 citation statements)

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“…Validity of Hill's equation in such a RVE was discussed in a number of papers: by Hill (1963Hill ( , 1967, by Havner (1971) Kreher and Pompe (1989), from the point of view of plastic deformation by Majumdar and McLaughlin (1975), and by Kafka (1972Kafka ( , 1983 Fig.l, i.e. in the case with a gradient of macroscopic stress.…”

Section: Equation (14)mentioning

confidence: 99%

“…approach by Kreher and Pompe (1989), from the point of view of plastic deformation by Majumdar and McLaughlin (1975), and by Kafka (1972Kafka ( , 1983. It is not possible to go into detail, but the conclusions can be characterized so that Hill's equation is applicable if the characteristic dimensions of the microstructure are small enough in relation to the dimensions of the volume element in question.…”

Section: Vml Hill's Equationmentioning

confidence: 99%

“…The left-hand figure A shows the case of homogeneous loading of a sample where Hill's equation is exactly valid. Some macroscopic subvolume in this sample -called RVE (representative volume element) -has the same macroscopic stressing, but on its surface there are fluctuations of microstress, connected with the microstructure.Validity of Hill's equation in such a RVE was discussed in a number of papers: by Hill (1963Hill ( , 1967, by Havner (1971) Kreher and Pompe (1989), from the point of view of plastic deformation by Majumdar and McLaughlin (1975), and by Kafka (1972Kafka ( , 1983 Fig.l, i.e. in the case with a gradient of macroscopic stress.…”

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Back Matter

2000

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Equation (1.4):is often called Hill's equation, and as mentioned in Chapter I it can rigorously be proven for a volume loaded at its surface 5 either by displacements u,{S) = e^Xy or by tractions t,{S) =o^Hj, derived, respectively, from uniform overall strain e /y or stress (7,y (Kafka 1983). However, the approximate validity of this equation is implicitly generally assumed for more general cases, as without it the use of experimental data, received with specimens of some material (data that specify the respective constitutive equations), for designs of constructions would not be possible (Kafka 1983). Hill himself anticipated the use of his equation generally for constitutive equations (not only for specimens with homogeneous boundary conditions), which is clear even from the title of one of his papers: 'The essential structure of constitutive laws for metal composites and polycrystals ' (Hill 1967). Nevertheless, the use of his equation for cases other than those with homogeneous boundary conditions is an approximation. In what follows the limits of such an applicability are discussed, the limits in which such an approximation is acceptable.In Fig.l two characteristic deviations from homogeneous boundary conditions are visualized. The left-hand figure A shows the case of homogeneous loading of a sample where Hill's equation is exactly valid. Some macroscopic subvolume in this sample -called RVE (representative volume element) -has the same macroscopic stressing, but on its surface there are fluctuations of microstress, connected with the microstructure.Validity of Hill's equation in such a RVE was discussed in a number of papers: by Hill (1963Hill ( , 1967, by Havner (1971) Kreher and Pompe (1989), from the point of view of plastic deformation by Majumdar and McLaughlin (1975), and by Kafka (1972Kafka ( , 1983 Fig.l, i.e. in the case with a gradient of macroscopic stress. It is assumed that for such cases Hill's equation is applicable in the investigated volume element if the variation of macroscopic stress due to the gradient is small in relation to the average macroscopic stress. Such an assumption is a natural application of the general principle saying that small deviations from the input data for which the investigated output is valid will cause small deviations from the validity of the output. Our assumption is corroborated also by the fact that constitutive equations that do not take into account gradients of stress are widely used and are able to describe real behavior of materials and constructions.Rigorous mathematicians sometimes question the use of Hill's equation, but rejection of it would imply rejection of the use of constitutive equations and their specification by experiments with material specimens. The use of Hill's equation is an approximation, but this approximation leads to results verified by experience. Appendix VIII.2 -Proof of Eq.(IU.29)Proof of the statement:and therefore j F{x) n 6V =0. v.and therefore F(x)"s0 and f(x) n = 1 q.e.d. 2), (n.2.4) and (n.2.7) are valid throughout the d...

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Section: Equation (14)mentioning

confidence: 99%

Section: Vml Hill's Equationmentioning

confidence: 99%

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2000

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“…Descriptions of effective-field methods occur in Kunin [10] and in Kreher and Pompe [11]. is, spheres to strongly oblate discs.…”

Section: Modelingmentioning

confidence: 99%

Void Shape in Sintered Titanium

Ledbetter

Dunn

Kim

et al. 1995

Review of Progress in Quantitative Nondestructive Evaluation

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“…In order to understand the micromechanical behaviour in more detail, the local microstress distribution must be considered. The average stresses in the platelets and the matrix can be described with a statistical approach [9], from which we get, for the isotropic part of the stress tensor, where v , K , G are the volume fraction, bulk modulus and shear modulus respectively, and AT is the temperature difference on going from the stress-free state to room temperature. The shape influence on the local residual stress field can be modeled by applying a generalization of the Eshelby approach, as shown by Kreher and Janssen [lo].…”

Section: Residual Stresses In Platelet Reinforced Ceramicsmentioning

confidence: 99%