The Generalized J-Integral: Symmetric Formulation, Relations to Stroh’s Complex Potentials, and Phonon–Phason Stress Intensity Factors
- 1Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Naukova Str. 3-b, 79060 Lviv, Ukraine
- 2Department of Computer Science and Cybersecurity, Lesya Ukrainka Volyn National University, Voli Avenue 13, 43025 Lutsk, Ukraine
Abstract
A unified analytical relation between the generalized Rice–Cherepanov J-integral and Stroh’s complex potentials is established for anisotropic elastic and quasicrystalline materials. A symmetric representation of both components of the J-integral is first derived in terms of the displacement and stress-function vectors. This representation treats the two components on an equal footing and provides a direct component-wise connection with the Stroh complex potentials. Explicit relations between both components of the J-integral and the Stroh complex potentials are established, extending the analysis beyond a representation in terms of a single complex configurational-force quantity. The resulting complex representation is then extended to quasicrystalline solids by incorporating the coupled phonon and phason fields into the generalized Stroh formalism. The asymptotic behavior of the complex potentials near a mathematical cut is used to derive explicit expressions for both components of the J-integral in terms of generalized phonon–phason stress intensity factors. In addition to the conventional intensity factors associated with displacement jumps, the formulation incorporates factors associated with stress jumps across the discontinuity. These stress-jump factors are essential for describing discontinuities subject to additional constraints, such as partially or completely blocked phason displacements at the faces of a quasicrystalline crack. The first component of the J-integral is expressed through the Barnett–Lothe tensors and contains a positive-definite contribution from the conventional intensity factors and a negative-definite quadratic contribution associated with the stress-jump factors. Consequently, a nonzero stress-jump contribution, which generally arises when the phason field is partially or completely blocked on the faces of a quasicrystalline crack, can reduce the J-integral. The overall effect of the phason constraint also depends on a mixed term coupling the displacement-jump and stress-jump intensity factors. The second component is expressed through the Stroh matrices and the Barnett–Lothe tensors and retains the corresponding mixed and stress-jump contributions. The derived relations recover previously established results for classical cracks and one-dimensional quasicrystals as limiting cases. The formulation provides a unified analytical framework for configurational forces associated with cracks, deformable and rigid inclusions, and discontinuities with constrained phason fields.
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BibTeX
@article{pasternak2026generalized,
title = {{The Generalized J-Integral: Symmetric Formulation, Relations to Stroh’s Complex Potentials, and Phonon–Phason Stress Intensity Factors}},
author = {Viktoriya Pasternak and Heorhiy Sulym and Iaroslav Pasternak},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214073},
url = {https://www.mdpi.com/2227-7390/14/21/4073},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The Generalized J-Integral: Symmetric Formulation, Relations to Stroh’s Complex Potentials, and Phonon–Phason Stress Intensity Factors
AU - Viktoriya Pasternak
AU - Heorhiy Sulym
AU - Iaroslav Pasternak
PY - 2026
DA - 2026/11/06
JO - Mathematics
DO - 10.3390/math14214073
UR - https://www.mdpi.com/2227-7390/14/21/4073
AB - A unified analytical relation between the generalized Rice–Cherepanov J-integral and Stroh’s complex potentials is established for anisotropic elastic and quasicrystalline materials. A symmetric representation of both components of the J-integral is first derived in terms of the displacement and stress-function vectors. This representation treats the two components on an equal footing and provides a direct component-wise connection with the Stroh complex potentials. Explicit relations between both components of the J-integral and the Stroh complex potentials are established, extending the analysis beyond a representation in terms of a single complex configurational-force quantity. The resulting complex representation is then extended to quasicrystalline solids by incorporating the coupled phonon and phason fields into the generalized Stroh formalism. The asymptotic behavior of the complex potentials near a mathematical cut is used to derive explicit expressions for both components of the J-integral in terms of generalized phonon–phason stress intensity factors. In addition to the conventional intensity factors associated with displacement jumps, the formulation incorporates factors associated with stress jumps across the discontinuity. These stress-jump factors are essential for describing discontinuities subject to additional constraints, such as partially or completely blocked phason displacements at the faces of a quasicrystalline crack. The first component of the J-integral is expressed through the Barnett–Lothe tensors and contains a positive-definite contribution from the conventional intensity factors and a negative-definite quadratic contribution associated with the stress-jump factors. Consequently, a nonzero stress-jump contribution, which generally arises when the phason field is partially or completely blocked on the faces of a quasicrystalline crack, can reduce the J-integral. The overall effect of the phason constraint also depends on a mixed term coupling the displacement-jump and stress-jump intensity factors. The second component is expressed through the Stroh matrices and the Barnett–Lothe tensors and retains the corresponding mixed and stress-jump contributions. The derived relations recover previously established results for classical cracks and one-dimensional quasicrystals as limiting cases. The formulation provides a unified analytical framework for configurational forces associated with cracks, deformable and rigid inclusions, and discontinuities with constrained phason fields.
SN - 2227-7390
ER -