International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 140-141

Section 2.5.4.3.1. Stress and strain relation

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

2.5.4.3.1. Stress and strain relation

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Stress is a measure of the deforming force applied to a solid per unit area. The stress on an elemental volume in the sample coordinates S1, S2, S3 contains nine components, given by[{\sigma _{ij}} = \left [{\matrix{ {{\sigma _{11}}} & {{\sigma _{12}}} & {{\sigma _{13}}} \cr {{\sigma _{21}}} & {{\sigma _{22}}} & {{\sigma _{23}}} \cr {{\sigma _{31}}} & {{\sigma _{32}}} & {{\sigma _{33}}} \cr } } \right].\eqno(2.5.63)]

A component is normal stress when the two indices are identical, or shear stress when the two indices differ. The group of the nine stress components is called the stress tensor. The stress tensor is a tensor of the second order. Under equilibrium conditions, the shear components must maintain the following relations:[{\sigma _{12}} = {\sigma _{21}},\ {\sigma _{23}} = {\sigma _{32}} \ {\rm and}\ {\sigma _{31}} = {\sigma _{13}}.\eqno(2.5.64)]Therefore, only six independent components define the stress state in a solid. The following stress states are typically measured:

  • Uniaxial: all stress components are zero except one normal stress component.

  • Biaxial: all nonzero components are within the S1S2 plane.

  • Biaxial with shear: [{\sigma _{33}} = 0], all other components are not necessarily zero.

  • Equibiaxial: a special case of biaxial stress where σ11 = σ22 = σ.

  • Triaxial: all components are not necessarily zero.

  • Equitriaxial: a special case of triaxial stress where σ11 = σ22 = σ33 = σ.

Strain is a measure of the resulting deformation of a solid body caused by stress. Strain is calculated from the change in the size and shape of the deformed solid due to stress. Analogous to normal stresses and shear stresses are normal strains and shear strains. The normal strain is calculated from the change in length of the solid body along the corresponding normal stress direction. Like the stress tensor, the strain tensor contains nine components:[{\varepsilon _{ij}} = \left [{\matrix{ {{\varepsilon _{11}}} & {{\varepsilon _{12}}} & {{\varepsilon _{13}}} \cr {{\varepsilon _{21}}} & {{\varepsilon _{22}}} & {{\varepsilon _{23}}} \cr {{\varepsilon _{31}}} & {{\varepsilon _{32}}} & {{\varepsilon _{33}}} \cr } } \right].\eqno(2.5.65)]The directions of all strain components are defined in the same way as for the stress tensor. Similarly, there are six independent components in the strain tensor. Strictly speaking, X-ray diffraction does not measure stresses directly, but strains. The stresses are calculated from the measured strains based on the elasticity of the materials. The stress–strain relations are given by the generalized form of Hooke's law:[{\sigma _{ij}} = {C_{ijkl}}{\varepsilon _{kl}},\eqno(2.5.66)]where [C_{ijkl}] are elastic stiffness coefficients. The stress–strain relations can also be expressed as[{\varepsilon _{ij}} = {S_{ijkl}}{\sigma _{kl}},\eqno(2.5.67)]where [S_{ijkl}] are the elastic compliances. For most polycrystalline materials without texture or with weak texture, it is practical and reasonable to consider the elastic behaviour to be isotropic and the structure to be homogeneous on a macroscopic scale. In these cases, the stress–strain relationship takes a much simpler form. Therefore, the Young's modulus E and Poisson's ratio ν are sufficient to describe the stress and strain relations for homogeneous isotropic materials:[\eqalignno{ & {\varepsilon _{11}} = {1 \over E}[{\sigma _{11}} - \nu ({\sigma _{22}} + {\sigma _{33}})], \cr & {\varepsilon _{22}} = {1 \over E}[{\sigma _{22}} - \nu ({\sigma _{33}} + {\sigma _{11}})], \cr & {\varepsilon _{33}} = {1 \over E}[{\sigma _{33}} - \nu ({\sigma _{11}} + {\sigma _{22}})] ,\cr & {\varepsilon _{12}} = {{1 + \nu } \over E}{\sigma _{12}},\quad {\varepsilon _{23}} = {{1 + \nu } \over E}{\sigma _{23}},\quad {\varepsilon _{31}} = {{1 + \nu } \over E}{\sigma _{31}}.&(2.5.68)} ]It is customary in the field of stress measurement by X-ray diffraction to use another set of macroscopic elastic constants, S1 and ½S2, which are given by[{\textstyle{1 \over 2}}S_2 = (1 + \nu)/E \ \,{\rm and}\ \, S_1 = - \nu /E.\eqno(2.5.69)]

Although polycrystalline materials on a macroscopic level can be considered isotropic, residual stress measurement by X-ray diffraction is done by measuring the strain in a specific crystal orientation of the crystallites that satisfies the Bragg condition. The stresses measured from diffracting crystallographic planes may have different values because of their elastic anisotropy. In such cases, the macroscopic elasticity constants should be replaced by a set of crystallographic plane-specific elasticity constants, [S_1^{\{hkl\}}] and [{\textstyle{1 \over 2}}S_2^{\left\{hkl\right\}}], called X-ray elastic constants (XECs). XECs for many materials can be found in the literature, measured or calculated from microscopic elasticity constants (Lu, 1996[link]). In the case of materials with cubic crystal symmetry, the equations for calculating the XECs from the macroscopic elasticity constants ½S2 and S1 are[\eqalignno{{\textstyle{{{1}} \over {{2}}}}{{S}}_{{2}}^{{{\{ hkl\} }}} &= {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}}{{[1}} + 3(0.2 - {{\Gamma (hkl))\Delta]}} &\cr {{S}}_{{1}}^{{{\{ hkl\} }}} &= {{{S}}_{{1}}} - {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}}[0.2 - {{\Gamma (hkl)]\Delta }},&(2.5.70)}]where[{{\Gamma (hkl)}} = {{{{{h}}^{{2}}}{{{k}}^{{2}}} + {{{k}}^{{2}}}{{{l}}^{{2}}} + {{{l}}^{{2}}}{{{h}}^{{2}}}} \over {{{{{(}}{{{h}}^{{2}}} + {{{k}}^{{2}}} + {{{l}}^{{2}}}{{)}}}^{{2}}}}} \ {\rm and}\ {{\Delta }} = {{{{5(}}{{{A}}_{{\rm{RX}}}} - {{1)}}} \over {{{3}} + {{2}}{{{A}}_{{\rm{RX}}}}}}.]In the equations for stress measurement hereafter, either the macroscopic elasticity constants ½S2 and S1 or the XECs [{{S}}_{{1}}^{\left\{ {{{hkl}}} \right\}}] and [{\textstyle{{{1}} \over {{2}}}}{{S}}_{{2}}^{\left\{ {{{hkl}}} \right\}}] are used in the expression, but either set of elastic constants can be used depending on the requirements of the application. The factor of anisotropy (ARX) is a measure of the elastic anisotropy of a material (He, 2009[link]).

References

He, B. B. (2009). Two-dimensional X-ray Diffraction. New York: John Wiley & Sons.Google Scholar
Lu, J. (1996). Handbook of Measurement of Residual Stress. Society for Experimental Mechanics. Lilburn: The Fairmont Press.Google Scholar








































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