15.19. Energies

Energies are available in the solution printout (by setting Item = VENG on the OUTPR command) or in postprocessing (by choosing items SENE, TENE, KENE, and AENE on the ETABLE command or using the PRENERGY command). For each element,

(15–249)
(15–250)
(15–251)

where:

NINT = number of integration points
{σ} = stress vector
el} = elastic strain vector
voli = volume of integration point i
  = plastic strain energy
Es = stress stiffening energy
 
[Ke] = element stiffness/conductivity matrix
[Se] = element stress stiffness matrix
{u} = element DOF vector
  = time derivative of element DOF vector
[Me] = element mass matrix
NCS = total number of converged substeps
{γ} = hourglass strain energy defined in Flanagan and Belytschko([242]) due to one point integrations.
[Q] = hourglass control stiffness defined in Flanagan and Belytschko([242]).

As may be seen from the bottom part of Equation 15–249 as well as Equation 15–250, all types of DOFs are combined, e.g., SOLID5 using both UX, UY, UZ, TEMP, VOLT, and MAG DOF. An exception to this is the piezoelectric elements, described in Piezoelectrics, which do report energies by separate types of DOFs in the NMISC record of element results. See Eigenvalue and Eigenvector Extraction when complex frequencies are used. Also, if the bottom part of Equation 15–249 is used, any nonlinearities are ignored. Elements with other incomplete aspects with respect to energy are reported in Table 15.2: Exceptions for Element Energies.

Artificial energy has no physical meaning. It is used to control the hourglass mode introduced by reduced integration. The rule-of-thumb to check if the element is stable or not due to the use of reduced integration is if   < 5% is true. When this inequality is true, the element using reduced integration is considered stable (i.e., functions the same way as fully integrated element).

Element type limitations for energy computation are given in Table 15.2: Exceptions for Element Energies.

Table 15.2  Exceptions for Element Energies

ElementException
FLUID29No potential energy
FLUID30No potential energy
LINK31No potential energy
LINK34No potential energy
COMBIN39No potential energy
SHELL41Foundation stiffness effects not included
SHELL61Thermal effects not included
FLUID141No potential energy
FLUID142No potential energy
  1. Warping implies for example that temperatures T1 + T3 ≠ T2 + T4, i.e., some thermal strain is locked in.


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