  As one of the most important topics studied in creep fracture mechanics,
mechanics fields at three-dimensional (3D) sharp V-notches and crack tip have
drawn tremendous attentions. With many years efforts on constraint theory
developed in creeping solids, there still seems dense fog on how in-plane and
out-of-plane constraint effects are interacted for 3D sharp V-notch and crack
in creeping solids. To shed lights on this topic, a 3D higher-order termed
solution for sharp V-notches in creeping materials subjected to mode 1 loading
is established by introducing the out-of-plane factor, which is the
out-of-plane stress divided by the sum of in-plane normal stress. The solution
can naturally be degenerated to a 3D crack. Based on the 3D higher-order term
solution, a new fracture parameter is proposed and combined with to
characterize 3D constraint effect. It is found that the stress exponents and
angular distribution of higher-order term for 3D notches and cracks are highly
related to . The proposed higher order termed solutions show better agreement
with the FEA results than the 3D leading-term and 2D two-term solutions,
especially for smaller notch angles and ligament width. Moreover, the presented
3D constraint theory shows that effects of and are highly interlinked rather
than simply separated. It implies that the 3D constraint level may be
significantly influenced by . The 3D mathematical solutions discussed in this
paper could enhance the understanding of the 3D effect and has the potential to
explain the 3D constraint effect on the notches and cracks under creep
conditions.
