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Linear elasticity
International Tables for Crystallography (2013). Vol. D, Section 1.3.3, pp. 80-85 [ doi:10.1107/97809553602060000902 ]
... 1.3.3. Linear elasticity 1.3.3.1. Hooke's law | | Let us consider a metallic bar of length loaded in pure tension (Fig. 1.3.3.1 ... the applied stress the bar resumes its original form. To a first approximation, the curve is linear, so that one can ... actually anharmonic and Hooke's law is only an approximation: a Taylor expansion up to the first term. A rigorous ...
Isotropic materials
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.5, pp. 84-85 [ doi:10.1107/97809553602060000902 ]
Isotropic materials 1.3.3.5. Isotropic materials The isotropy relation between elastic compliances and elastic stiffnesses is given in Section 1.3.3.2.3. For reasons of symmetry, the directions of the eigenvectors of the stress and strain tensors are necessarily the same in an isotropic medium. If we take these directions as axes, the two ...
Variation of Young's modulus with orientation
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.4, p. 84 [ doi:10.1107/97809553602060000902 ]
... sufficient to change the axes of the tensor . If A is the matrix associated with the change of axes, leading ... zinc (Fig. 1.3.3.4d, ), the surface is of revolution and has a larger symmetry. It is interesting to compare the differences in ... the crystal is pseudo-isotropic and the surface is practically a sphere. Figure 1.3.3.4 | | Representation surface of the inverse of ...
Young's modulus, Poisson's ratio
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.3, p. 83 [ doi:10.1107/97809553602060000902 ]
... modulus, Poisson's ratio If the applied stress reduces to a uniaxial stress, , the strain tensor is of the form In ... that Young's modulus (equation 1.3.3.1) is The elongation of a bar under the action of a uniaxial stress is characterized by and the diminution of ...
Linear compressibility
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.2, p. 83 [ doi:10.1107/97809553602060000902 ]
Linear compressibility 1.3.3.4.2. Linear compressibility Under the action of a hydrostatic pressure, each vector assumes a different elongation. This elongation is given by equation (1.3.1.6): where ... for the coefficient of linear compressibility In the case of a cubic or isotropic medium, this expression reduces to The ...
Volume compressibility
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.1, p. 83 [ doi:10.1107/97809553602060000902 ]
Volume compressibility 1.3.3.4.1. Volume compressibility Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the ...
Particular elastic constants
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4, pp. 83-84 [ doi:10.1107/97809553602060000902 ]
... 1.3.3.4. Particular elastic constants 1.3.3.4.1. Volume compressibility | | Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the ...
Elastic strain energy
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.3, pp. 82-83 [ doi:10.1107/97809553602060000902 ]
... 1.3.2.7) of the strain energy stored per unit volume in a medium for a small deformation can be integrated when the medium is strained under a stress according to linear elasticity. Applying relation (1.3.3.2), one ...
Passage from elastic compliances to elastic stiffnesses
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.2.3, p. 82 [ doi:10.1107/97809553602060000902 ]
Passage from elastic compliances to elastic stiffnesses 1.3.3.2.3. Passage from elastic compliances to elastic stiffnesses We have noted already that the matrix is the inverse of the matrix . These matrices can be written for cubic and isotropic materials as follows: where we have, for isotropic materials, We easily find that ...
Matrix notation - reduction of the number of independent components
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.2.2, pp. 81-82 [ doi:10.1107/97809553602060000902 ]
... of dummy indices ij and lk: Since the energy is a state function with a perfect differential, one can interchange the order of the differentiations ... cannot apply to them the usual rules of transformation under a change of base since they are only valid for ...
International Tables for Crystallography (2013). Vol. D, Section 1.3.3, pp. 80-85 [ doi:10.1107/97809553602060000902 ]
... 1.3.3. Linear elasticity 1.3.3.1. Hooke's law | | Let us consider a metallic bar of length loaded in pure tension (Fig. 1.3.3.1 ... the applied stress the bar resumes its original form. To a first approximation, the curve is linear, so that one can ... actually anharmonic and Hooke's law is only an approximation: a Taylor expansion up to the first term. A rigorous ...
Isotropic materials
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.5, pp. 84-85 [ doi:10.1107/97809553602060000902 ]
Isotropic materials 1.3.3.5. Isotropic materials The isotropy relation between elastic compliances and elastic stiffnesses is given in Section 1.3.3.2.3. For reasons of symmetry, the directions of the eigenvectors of the stress and strain tensors are necessarily the same in an isotropic medium. If we take these directions as axes, the two ...
Variation of Young's modulus with orientation
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.4, p. 84 [ doi:10.1107/97809553602060000902 ]
... sufficient to change the axes of the tensor . If A is the matrix associated with the change of axes, leading ... zinc (Fig. 1.3.3.4d, ), the surface is of revolution and has a larger symmetry. It is interesting to compare the differences in ... the crystal is pseudo-isotropic and the surface is practically a sphere. Figure 1.3.3.4 | | Representation surface of the inverse of ...
Young's modulus, Poisson's ratio
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.3, p. 83 [ doi:10.1107/97809553602060000902 ]
... modulus, Poisson's ratio If the applied stress reduces to a uniaxial stress, , the strain tensor is of the form In ... that Young's modulus (equation 1.3.3.1) is The elongation of a bar under the action of a uniaxial stress is characterized by and the diminution of ...
Linear compressibility
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.2, p. 83 [ doi:10.1107/97809553602060000902 ]
Linear compressibility 1.3.3.4.2. Linear compressibility Under the action of a hydrostatic pressure, each vector assumes a different elongation. This elongation is given by equation (1.3.1.6): where ... for the coefficient of linear compressibility In the case of a cubic or isotropic medium, this expression reduces to The ...
Volume compressibility
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4.1, p. 83 [ doi:10.1107/97809553602060000902 ]
Volume compressibility 1.3.3.4.1. Volume compressibility Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the ...
Particular elastic constants
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.4, pp. 83-84 [ doi:10.1107/97809553602060000902 ]
... 1.3.3.4. Particular elastic constants 1.3.3.4.1. Volume compressibility | | Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the ...
Elastic strain energy
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.3, pp. 82-83 [ doi:10.1107/97809553602060000902 ]
... 1.3.2.7) of the strain energy stored per unit volume in a medium for a small deformation can be integrated when the medium is strained under a stress according to linear elasticity. Applying relation (1.3.3.2), one ...
Passage from elastic compliances to elastic stiffnesses
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.2.3, p. 82 [ doi:10.1107/97809553602060000902 ]
Passage from elastic compliances to elastic stiffnesses 1.3.3.2.3. Passage from elastic compliances to elastic stiffnesses We have noted already that the matrix is the inverse of the matrix . These matrices can be written for cubic and isotropic materials as follows: where we have, for isotropic materials, We easily find that ...
Matrix notation - reduction of the number of independent components
International Tables for Crystallography (2013). Vol. D, Section 1.3.3.2.2, pp. 81-82 [ doi:10.1107/97809553602060000902 ]
... of dummy indices ij and lk: Since the energy is a state function with a perfect differential, one can interchange the order of the differentiations ... cannot apply to them the usual rules of transformation under a change of base since they are only valid for ...
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