Abstract
In this article, an alternative to the classical dynamic equation formulation is presented. To achieve this goal, we need to derive the reciprocal theorem in rates and the principle of virtual work in rates, in a small deformation regime, with which we will be able to obtain an expression for damping force. In this new formulation, some terms that are not commonly considered in the classical formulation appear, e.g., the term that is function of jerk (the rate of change of acceleration). Moreover, in this formulation the term that characterizes material nonlinearity, in dynamic analysis, appears naturally.
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Abbreviations
- B :
-
Continuum medium
- \({[{\bf{B}}]}\) :
-
Derivative of shape function matrix
- \({\vec{{\bf{b}}}}\) :
-
Body force (per unit mass)
- \({{\bf{C}}^{e}}\) :
-
Elasticity tensor (fourth-order tensor)
- \({[{\it{\bf{C}}}}]\) :
-
Elasticity matrix (Voigt notation)
- \({[{\bf{D}}]}\) :
-
Damping matrix
- dV :
-
Volume element
- \({\{{\bf d}\}}\) :
-
Nodal displacement vector
- \({\{{\dot{\bf d}}\}}\) :
-
Nodal velocity vector
- \({\{{\ddot{\bf d}}\}}\) :
-
Nodal acceleration vector
- \({\{\dddot{\bf d}\}}\) :
-
Nodal jerk vector
- \({\{\it{\bf{F}}^{({\rm Damp})}\}}\) :
-
Nodal damping force vector
- \({\{\it{\bf{F}}^{(B)}\}}\) :
-
Nodal body force vector
- \({\{\it{\bf{F}}^{(S)}\}}\) :
-
Nodal surface force vector
- \({[{\bf{G}}^{\vec{{\bf{t}}}}]}\) :
-
Interpolation matrix for the traction vector
- k :
-
Elastic constant of spring
- \({[{\bf{K}}]}\) :
-
Stiffness matrix
- \({[{\bf{M}}]}\) :
-
Mass matrix
- \({\hat{{\bf{n}}}}\) :
-
Outward unit normal to the boundary
- \({[{\bf{N}}]}\) :
-
Shape function matrix
- \({[{\bf{N}}^{\vec{{\bf{t}}}}]}\) :
-
Shape function for traction vector
- S :
-
Boundary of B
- \({\vec{\bf{t}}{}^{(\hat{{\bf{n}}})}}\) :
-
Traction vector
- t :
-
Time; \({\Delta t}\) —time increment
- \({\vec{{\bf{u}}}}\) :
-
Displacement field
- \({\dot{\vec{{\bf{u}}}}}\) :
-
Velocity field
- \({\ddot{\vec{{\bf{u}}}}}\) :
-
Acceleration field
- \({{\dddot{\vec{{\bf{u}}}}}}\) :
-
Jerk field
- \({\vec{{\bf{\it{x}}}}}\) :
-
Vector position
- \({\alpha, \xi }\) :
-
Damping coefficients
- \({\boldsymbol{\varepsilon} }\) :
-
Infinitesimal strain tensor (2nd order tensor)
- \({\rho }\) :
-
Mass density
- \({\boldsymbol{\sigma}}\) :
-
Cauchy stress tensor (2nd order tensor)
- \({\varPsi ^{e}}\) :
-
Strain energy density
- \({({\bullet})^{\rm T}}\) :
-
Transpose of \({({\bullet})}\)
- \({({\bullet})^{-1}}\) :
-
Inverse of \({({\bullet})}\)
- \({({\bullet})^{\rm sym}}\) :
-
Symmetric part of \({({\bullet})}\)
- \({\cdot }\) :
-
Scalar product
- \({\boldsymbol{:}}\) :
-
Double scalar product
- \({\frac{{D}({\bullet})}{{D}t}\equiv \dot {{\bullet}}}\) :
-
Material time derivative of \({\left( {\bullet}\right)}\)
- \({\nabla {\bullet}\equiv {\bf{grad}}({\bullet})}\) :
-
Gradient of \({{\bullet}}\)
- \({\nabla \cdot {\bullet}\equiv {\bf{div}}({\bullet})}\) :
-
Divergence of \({{\bullet}}\)
- \({\vec {{\bullet}}}\) :
-
Vector (tensorial notation)
- \({\hat{{{\bullet}}}}\) :
-
Unit vector (tensorial notation)
- \({[{\bullet}]}\) :
-
Matrix (Voigt notation)
- \({\{{\bullet}\}}\) :
-
Vector (Voigt notation)
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Communicated by Andreas Öchsner.
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Chaves, E.W.V. Dynamic analysis: a new point of view. Continuum Mech. Thermodyn. 28, 853–868 (2016). https://doi.org/10.1007/s00161-015-0419-4
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DOI: https://doi.org/10.1007/s00161-015-0419-4