Abstract
This chapter is about the role of the dissipation rate function—and other functions derived from it—in determining the constitutive behaviour of dissipative materials. It consists of a discussion of some general theory, followed by examples. We address the class of materials for which knowledge of the functional form of the dissipation rate function supplies the complete constitutive response, without recourse to further assumptions. A careful distinction is drawn between functions that are true potentials and those that are pseudopotentials (defined in the chapter), in order to clarify some aspects of terminology that the Author has elsewhere found confusing. In plasticity theory the intimate relationship between the dissipation rate function and the yield surface is explored. The chapter is illustrated by examples of simple one- and two-dimensional conceptual models, as well as full continuum models. Both rate independent (plastic) and rate-dependent (viscous, or viscoplastic) models are addressed.
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Abbreviations
- \( d\left( {x,v} \right) \) :
-
Dissipation rate function
- \( d^{*}\left( {x,\chi } \right) \) :
-
Dissipation rate function (alternative functional form)
- \( I_{\rm X} \left( \chi \right) \) :
-
Indicator function of a set X
- k :
-
Plastic strength
- n :
-
Order of homogeneous dissipation rate function in velocities
- \( N_{\rm X} \left( \chi \right) \) :
-
Normal cone to a convex set X
- r :
-
Reference velocity
- R :
-
Reference force
- v :
-
Generalised velocity (of state variable)
- x :
-
State variable
- \( w\left( {x,\chi } \right) \) :
-
Flow potential
- \( y\left( {x,\chi } \right) \) :
-
Yield function
- \( \bar{y}\left( {x,\chi } \right) \) :
-
Canonical form of yield function
- \( z\left( {x,v} \right) \) :
-
Force potential
- \( \gamma_{\rm X} \left( \chi \right) \) :
-
Gauge or Minkowski function of a convex set X
- \( \uptheta \) :
-
Absolute temperature
- \( \lambda \) :
-
Scalar multiplier
- µ:
-
Viscosity
- \( \Uplambda \) :
-
Lagrangian multiplier
- \( \uptau \) :
-
Dummy variable in integral transform
- \( \chi \) :
-
Generalized force
- \( S\left( x \right) = \left\{ {\begin{array}{*{20}l} { - 1,} & {x < 0} \\ { \in \left[ { - 1,1} \right],} & {x = 0} \\ { + 1,} & {x > 0} \\ \end{array} } \right. \) :
-
Generalised signum function
- \( \left\langle x \right\rangle = \left\{ {\begin{array}{*{20}l} {0,} & {x < 0} \\ {x,} & {x \ge 0} \\ \end{array} } \right. \) :
-
Macaulay brackets
- \( \left\langle {x,y} \right\rangle \) :
-
Inner product
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Houlsby, G.T. (2014). Dissipation Rate Functions, Pseudopotentials, Potentials and Yield Surfaces. In: Dewar, R., Lineweaver, C., Niven, R., Regenauer-Lieb, K. (eds) Beyond the Second Law. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40154-1_4
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DOI: https://doi.org/10.1007/978-3-642-40154-1_4
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