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Methods of Analysis I: Review of Vectors, Dyadics, Matrices, and Determinants
Published in Ronald L. Huston, Principles of Biomechanics, 2008
The dyadic product of two vectors is simply the multiplication of the vectors following the usual rules of algebra except the commutative rule. That is, () AB≠BA
Implicit non-ordinary state-based peridynamics model for linear piezoelectricity
Published in Mechanics of Advanced Materials and Structures, 2022
Francisco S. Vieira, Aurélio L. Araújo
Having the equation of motion, it is important to know how to express the force density vector state in terms of the material properties of the problem. In non-ordinary state-based peridynamics, a class of materials that can be used are the correspondence material models. These have the particularity of establishing a bridge between classical continuum mechanics and peridynamics. As derived by Silling [24] the force density vector state can be expressed as a function of the first Piola-Kirchhoff stress tensor P as where B is the shape tensor and ω is the weight function. The shape tensor is given by where the operator ’⊗’ denotes the dyadic product between two vectors. Additionally, a nonlocal PD deformation gradient F is also defined as
Modeling stress–strain response of shape memory alloys during reorientation of self-accommodated martensites with different morphologies
Published in Mechanics of Advanced Materials and Structures, 2022
Mahendaran Uchimali, Srikanth Vedantam
In order to obtain the form of consider the triangle in reference and current configurations as shown as inset in Figure 1. The deformation of the triangle can be described by the relative position vectors and in reference and current configurations, respectively. We assume a linear transformation function for each Delaunay triangle that maps the relative position vectors r and d such that and This linear transformation is referred as the discrete deformation tensor. By solving the pair of equations for each triangle, the discrete deformation tensor is expressed as a function of particle positions where represents the dyadic product between the two vectors a and b.