For a material in linear elastic regime, the work density is given as
0.5 * σ:ε ( double dot product of stress tensor and strain tensor)
Say, a material is undergoing plastic deformation, also locally the stress state is multi axial.
How would the definition of the total work density for this case ?
Can we use the energy conjugates like PK2 :E ( PK2 is the second Piola - Kirchhoff tensor, E is the Green strain tensor) ?
Hello, Prithivirajan! In my research (DOI: 10.1007/s11003-015-9789-9) I’v calculated the total work energy density or strain energy density that is the same stuff as a sum of three components of energy that are spent on deformation of material in three main directions. Each one of this components is calculated as an aria bounded by the stress-strain curve for certain direction and the strain axis. Formulas you can find in the article (sorry for pure English translation). Hope it will help you.
Hello. Indeed, you can use conjugate stress and strain measures to calculate the density of power which is necessary to deform a piece of material. More specifically, a each time t, the application of a deformation increment to a volume element requires a a total energy increment dw or dW whose expression is:
dw = sigma : d (where sigma is the Cauchy stress and d is the eulerian strain rate)
dW = P:dF (where P is first Piola-Kirchoff stress tensor and F is the deformation gradient)
dW = S:dE (where S is the second Piola-Kirchoff stress tensor and E is the Green-Lagrange strain tensor).
With the above notations, dw is the total energy increment per volume unit in the deformed configuration. dW is the total energy increment per volume unit in the reference configuration. If you want to know the total amount of energy, you just need to integrate dw or dW over the entire loading path. Also, the above relations are valid for any type of constitutive behavior (e.g. elasticity, plasticity, elasto-plasticity...) and any stress state.
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