Stress strain tensors with their application to x-ray stress measurement

This paper describes in detail the method of obtaining the formulas of stress-strain tensor that express the directional dependence of stress-strain, that is, how these values change in response to coordinate transformation, and clarifies the preconditions for supporting both formulas. The two conversion formulas are both the second order of tensor, and the formula of strain tensor not only does not use the relational expression of stress and strain at all, but also is obtained completely independently of the formula of stress tensor. Except for the condition that the strain is very small (elastic deformation) in the conversion formula of strain, both formulas unconditionally come into effect. In other words, both formulas hold true even in the isotropic elastic body or anisotropic elastic body. It was shown that the conversion formula of strain can be derived from the conversion formula of stress using the formula of Hooke for isotropic elastic body. From these three-dimensional expressions, the two-dimensional stress-strain coordinate conversion formula that is used for Mohr's stress-strain circle was derived. It was shown that these formulas hold true for three-dimensional stress condition with stress-strain components in the three-axial direction that are not plane stress nor plane strain condition. In addition, as an application case of this theory, two-dimensional and three-dimensional X-ray stress measurements that are effective for residual stress measurement were shown. (A.O.)