Thermal considerations

SDT-visc Contents Functions

2.5 Thermal considerations

In this section one addresses models needed to evaluate the steady state temperature field for a forced harmonic response.

2.5.1 Thermal model

xxx detail xxx

Exchange on free structure interfaces.

Exchange at the internal viscoelastic/metal interfaces.

2.5.2 Heat source due to viscoelastic behavior

Assuming a forced harmonic response, one has

(2.68)

where stress is related to strain through the complex modulus

(2.69)

or in the time domain

(2.70)

Integrating the power dissipated over a period of the forced response, one obtains the spatial distribution of dissipated power

(2.71)

Note that the maximum strain energy during the cycle is given by

(2.72)

which gives a simple way to estimate the loss factor (??) associated with the local stress/strain state.

Despite the local nature of dissipation, one may want to verify that the integral over the volume of the dissipated energy is equal to the input power. Noting H(ω) the transfer collocated with the input, the equivalent power input in the structure is given by

(2.73)

In general, there is no direct way to measure the maximum strain energy in a structure and thus no experimental definition of a global or system loss factor.

For models, the energy can be computed but there is not particular reason for the local energy (??) to reach a maximum value at all points simultaneously. One can thus search for the maximum strain energy in the system using

(2.74)

In the case of normal modes, the strain energy is equal to the square of the mode pulsation so that the MSE methods can be meaningful (see section ??).

An illustration of global loss factor use for uniform and non-uniform material loss can be found in in t_visco('ThermoPower')

2.5.3 Cantilever plate example

To perform coupled thermoelastic modes computation, 3 commands are available in comp12 :

Thermo1Assemble assembles structure matrices, thermal matrices and coupled thermoelastic matrices.
Thermo1Build builds MVR reduced model.
Thermo1Modes computes associated complex modes.

First a finite element model (mdl must be created, with element associated to structure properties (m_elastic and p_solid). Thermal properties (m_heat and p_heat) must be present in model material and element property stack.
Matrices must be assembled as following:
RO=comp12('Thermo1Assemble',mdl,RO);
RO is the coupling data structure with fields

.MatId [Matid_struct MatId_therm], first is the MatId of the structure material property, and second the thermal one.
.ProId [ProId_struct ProId_therm TO Matid_struct MatId_therm]. T0 is the thermal coupling (C, Inf if no coupling).

Then reduced MVR can be build using:
MVR=comp12('Thermo1Build-reduce',mdl,RO); If command option -reduce is given model is reduced according to real modeshapes. Then eig options must be given in RO.EigOpt.

Complex coupled thermoelastic modes can then be computed using
def=comp12('Thermo1Modes',MVR);

2.5.4 Thermo-elastic damping

To perform coupled thermoelastic modes computation, 3 commands are available in comp12 :

Thermo1Assemble assembles structure matrices, thermal matrices and coupled thermoelastic matrices.
Thermo1Build builds MVR reduced model.
Thermo1Modes computes associated complex modes.

.MatId [Matid_struct MatId_therm], first is the MatId of the structure material property, and second the thermal one.
.ProId [ProId_struct ProId_therm TO Matid_struct MatId_therm]. T0 is the thermal coupling (C, Inf if no coupling).

Complex coupled thermoelastic modes can then be computed using
def=comp12('Thermo1Modes',MVR);

Following full example can be found in comp12('numeric'). It illustrates the computation of thermo-elastic damping in a simple rectangular plate example.

 T0=20; % coupling temperature
 % Build model: - - -
 Nz=5;
 mdl=femesh(sprintf('testhexa8 divide 10 10 %i',Nz));
 h=2e-3;
 mdl.Node(:,5:7)=mdl.Node(:,5:7)*diag([.304 .192 h]);
 mdl.Elt=feutil('set groupall matid 1',mdl);
 mdl.Elt=feutil('set groupall proid 1',mdl);

 % define material properties: - - -
 mdl.pl=m_elastic('dbval 1 Aluminum');  % structure
 mdl=feutil('setmat 1 alpha=22e-6 T0=20',mdl);% XXX alpha at what temperature ? 
 mdl.pl=m_heat(mdl.pl,'dbval 2 Aluminum'); % therm
 % define element properties:
 mdl.il=p_solid('dbval 1 d3 -3'); % structure
 mdl.il=p_heat(mdl.il,'dbval 2 d3 -3'); % therm
 mdl=feutil('lin2quad',mdl);
 RO=struct; % build options: - - - 
 RO.MatId=[1 2]; % structure | therm 
 RO.ProId=[1 2   T0    1 2]; % structure | therm | Temperature(if coupling, Inf if not) Matid_struct MatId_therm

 % Assemble elementary matrices according to coupling defined in RO :
 RO=comp12('Thermo1Assemble',mdl,RO);

 % Build MVR with coupling :
 RO.EigOpt=[5 40 1e3];
 MVR=comp12('Thermo1Build-reduce',mdl,RO);

 % Compute associated complex modes:
 def=comp12('Thermo1Modes',MVR); % compute modes
 cf=feplot(mdl); cf.def=def;