Mathematical and Numerical Study of Symmetry and Positivity of the Tensor-Valued Spring Constant Defined by P1-FEM for Two- and Three-Dimensional Linear Elasticity

Abstract In this study, we consider a spring-block system that approximates a d -dimensional linear elastic body, where $$d=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> or $$d=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . We derive a $$d\times d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>×</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> matrix for the spring constant using the P1-finite element method (P1-FEM) with a triangular mesh for the linear elasticity equations. We mathematically analyze the symmetry and positive-definiteness of the spring constant. Even if we assume the full symmetry of the elasticity tensor, the symmetry of the matrix obtained for the spring constant is not trivial. However, we have succeeded in proving this in a unified manner for both two-dimensional and three-dimensional cases. This is an alternative proof of the two-dimensional case in Notsu-Kimura (2014) and is a new result for the three-dimensional case. We provide a necessary and sufficient condition for the spring constant to be positive-definite in the case of an isotropic elasticity tensor, along with a sufficient condition in terms of mesh regularity and Poisson’s ratio. These theoretical results are supported by several numerical experiments. The positive-definiteness of the spring constant derived from the FEM plays a vital role in fracture simulations of elastic bodies using the spring-block system.