This paper presents a specific network architecture for approximation of the first Piola-Kirchhoff stress.The neural network enables us to construct the constitutive relation based on both macroscopic observations and...This paper presents a specific network architecture for approximation of the first Piola-Kirchhoff stress.The neural network enables us to construct the constitutive relation based on both macroscopic observations and atomistic simulation data.In contrast to traditional deep learning models,this architecture is intrinsic symmetric,guarantees the frame-indifference and material-symmetry of stress.Specifically,we build the approximation network inspired by the Cauchy-Born rule and virial stress formula.Several numerical results and theory analyses are presented to illustrate the learnability and effectiveness of our network.展开更多
<正> With respect to small deformation gradient tensor,the displacement gradientmoduli Φ_(iLjM)~* at the reference system are introduced by taking into account the rela-tion of the first Piola-Kirchhoff stress ...<正> With respect to small deformation gradient tensor,the displacement gradientmoduli Φ_(iLjM)~* at the reference system are introduced by taking into account the rela-tion of the first Piola-Kirchhoff stress tensor T_(iL) and displacement gradient tensor u_(jM)in the paper.Then the specific expression of Φ_(iLjM) is derived by the comparison of theexpansion terms of Helmholtz free energy H.Finally,the coefficients (?)_(iLjM) related toΦ_(iLjM) are obtained by using ultrasonic measurement and the elastic tensor C_(iLjM) whichdepends on (?)_(iLjM) and initial stresses in the reference system are given.展开更多
Certain stress and strain form a thermodynamic conjugate pair such that their strain energy equals to a scalar-valued potential energy.Different atomistic stresses and strains are analytically derived based on the wor...Certain stress and strain form a thermodynamic conjugate pair such that their strain energy equals to a scalar-valued potential energy.Different atomistic stresses and strains are analytically derived based on the work conjugate relation.It is numerically verified with both two-body and three-body potentials that the atomistic Kirchhoff stress,first-order Piola–Kirchhoff stress and second-order Piola–Kirchhoff stress are conjugates to atomistic logarithmic strain,deformation gradient and Lagrangian strain,respectively.Virial stress at 0 K based on original volume is the special form of atomistic Kirchhoff stress for pair potential.It is numerically verified that Hencky strain is not conjugate to any stress.展开更多
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the National Nature Science Foundation of China(Nos.12125103 and 12071362)+1 种基金the Natural Science Foundation of Hubei Province(Nos.2021AAA010 and 2019CFA007)by the Fundamental Research Funds for the Central Universities.The numerical calculations have been done at the Super Computing Center of Wuhan University。
文摘This paper presents a specific network architecture for approximation of the first Piola-Kirchhoff stress.The neural network enables us to construct the constitutive relation based on both macroscopic observations and atomistic simulation data.In contrast to traditional deep learning models,this architecture is intrinsic symmetric,guarantees the frame-indifference and material-symmetry of stress.Specifically,we build the approximation network inspired by the Cauchy-Born rule and virial stress formula.Several numerical results and theory analyses are presented to illustrate the learnability and effectiveness of our network.
文摘<正> With respect to small deformation gradient tensor,the displacement gradientmoduli Φ_(iLjM)~* at the reference system are introduced by taking into account the rela-tion of the first Piola-Kirchhoff stress tensor T_(iL) and displacement gradient tensor u_(jM)in the paper.Then the specific expression of Φ_(iLjM) is derived by the comparison of theexpansion terms of Helmholtz free energy H.Finally,the coefficients (?)_(iLjM) related toΦ_(iLjM) are obtained by using ultrasonic measurement and the elastic tensor C_(iLjM) whichdepends on (?)_(iLjM) and initial stresses in the reference system are given.
文摘Certain stress and strain form a thermodynamic conjugate pair such that their strain energy equals to a scalar-valued potential energy.Different atomistic stresses and strains are analytically derived based on the work conjugate relation.It is numerically verified with both two-body and three-body potentials that the atomistic Kirchhoff stress,first-order Piola–Kirchhoff stress and second-order Piola–Kirchhoff stress are conjugates to atomistic logarithmic strain,deformation gradient and Lagrangian strain,respectively.Virial stress at 0 K based on original volume is the special form of atomistic Kirchhoff stress for pair potential.It is numerically verified that Hencky strain is not conjugate to any stress.
