Tables for
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, p. 142

Section Equations for various stress states

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Equations for various stress states

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The general triaxial stress state is not typically measured by X-ray diffraction because of low penetration. For most applications, the stresses in a very thin layer of material on the surface are measured by X-ray diffraction. It is reasonable to assume that the average normal stress in the surface-normal direction is zero within such a thin layer. Therefore, [{\sigma _{33}} = 0], and the stress tensor has five nonzero components. In some of the literature this stress state is denoted as triaxial. In order to distinguish this from the general triaxial stress state, here we name this stress state as the `biaxial stress state with shear'. In this case, we can obtain the linear equation for the biaxial stress state with shear:[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{\rm ph}}{\sigma _{\rm ph}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),\eqno(2.5.80)]where the coefficient [{{{p}}_{{{\rm ph}}}} = {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}} + {{3}}{{{S}}_{{1}}}] and σph is the pseudo-hydrostatic stress component introduced by the error in the stress-free d-spacing. In this case, the stresses can be measured without the accurate stress-free d-spacing, since this error is included in σph. The value of σph is considered as one of the unknowns to be determined by the linear system. With the measured stress-tensor components, the general normal stress (σϕ) and shear stress (τϕ) at any arbitrary angle ϕ can be given by[\eqalignno{\sigma_\varphi &=\sigma_{11}\cos^2\varphi +\sigma_{12}\sin 2\varphi +\sigma_{22}\sin^2\varphi,&(2.5.81)\cr \tau_\varphi &= \sigma_{13}\cos\varphi +\sigma_{23}\sin\varphi. &(2.5.82)}]

Equation (2.5.81)[link] can also be used for other stress states by removing the terms for stress components that are zero. For instance, in the biaxial stress state [{\sigma _{33}} = {\sigma _{13}} = {\sigma _{23}} = 0], so we have[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{\rm ph}}{\sigma _{\rm ph}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right).\eqno(2.5.83)]

In the 2D stress equations for any stress state with σ33 = 0, we can calculate stress with an approximation of do (or 2θo). Any error in do (or 2θo) will contribute only to a pseudo-hydrostatic term σph. The measured stresses are independent of the input do (or 2θo) values (He, 2003[link]). If we use [d'_0] to represent the initial input, then the true do (or 2θo) can be calculated from σph with[\eqalignno{d_0&=d_0^\prime \exp\left({{1-2\nu}\over E}\sigma_{\rm ph}\right),&(2.5.84)\cr \theta_0&=\arcsin\left[\sin\theta_0^\prime\exp\left({{1-2\nu}\over E}\sigma_{\rm ph}\right)\right].&(2.5.85)}]

Care must be taken that the σph value also includes the measurement error. If the purpose of the experiment is to determine the stress-free d-spacing do, the instrument should be first calibrated with a stress-free standard of a similar material.


He, B. B. (2003). Introduction to two-dimensional X-ray diffraction. Powder Diffr. 18, 71–85.Google Scholar

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