Effective potentials in nonlinear polycrystals and quadrature formulae

Area of Science:

• Materials Science
• Solid Mechanics
• Computational Mechanics

Background:

• Estimating effective properties of nonlinear polycrystals is challenging.
• Existing methods often rely on approximations that may not capture complex material behaviors.
• Accurate modeling is essential for predicting material performance under various conditions.

Purpose of the Study:

• To present a family of estimates for effective potentials in nonlinear polycrystals.
• To investigate quadrature formulae for expressing integrals of nonlinear functions in terms of field moments.
• To apply these formulae to estimate effective potentials in polycrystals using a reduced-order model.

Main Methods:

• Investigation of several quadrature formulae to approximate integrals of nonlinear functions of local fields.
• Application of these formulae to estimate effective potentials in polycrystals governed by two potentials.
• Utilizing a non-uniform transformation field analysis (NTFA) reduced-order model.

Main Results:

• Two quadrature formulae were found to reduce to known schemes, including a recent proposition.
• The proposed quadrature formulae improve upon the tangent second-order approximation in porous crystals at high stress triaxiality.
• Higher-order quadrature formulae are necessary for satisfactory accuracy in highly nonlinear porous crystals under high stress triaxiality.

Conclusions:

• The developed quadrature formulae offer improved estimates for effective potentials in nonlinear polycrystals.
• The choice of quadrature formula significantly impacts accuracy, especially under extreme conditions.
• Higher-order methods are recommended for precise modeling of nonlinear porous materials exhibiting high stress triaxiality.