The capability to do a thermoelastic analysis exists in the following elements:
PLANE222 - 2-D 4-Node Coupled-Field Solid |
PLANE223 - 2-D 8-Node Coupled-Field Solid |
SOLID226 - 3-D 20-Node Coupled-Field Solid |
SOLID227 - 3-D 10-Node Coupled-Field Solid |
These elements support both the thermal expansion and piezocaloric effects, and use the strong (matrix) coupling method.
In addition to the above, the following elements support the thermal expansion effect only in the form of a thermal strain load vector, i.e. use weak coupling method:
SOLID5 - 3-D 8-Node Coupled-Field Solid |
PLANE13 - 2-D 4-Node Coupled-Field Solid |
SOLID98 - 3-D 10-Node Coupled-Field Solid |
Constitutive Equations of Thermoelasticity
The coupled thermoelastic constitutive equations (Nye([360])) are:
(10–30) |
(10–31) |
where:
{ε} = total strain vector = [ε_{x} ε_{y} ε_{z} ε_{xy} ε_{yz} ε_{xz}]^{T} |
S = entropy density |
{σ} = stress vector = [σ_{x} σ_{y} σ_{z} σ_{xy} σ_{yz} σ_{xz}]^{T} |
ΔT = temperature change = T - T_{ref} |
T = current temperature |
T_{0} = absolute reference temperature = T_{ref} + T_{off} |
T_{ref} = reference temperature (input on TREF command or as REFT on MP command) |
T_{off} = offset temperature from absolute zero to zero (input on TOFFST command) |
[D] = elastic stiffness matrix (inverse defined in Equation 2–4 or input using TB,ANEL command) |
{α} = vector of coefficients of thermal expansion = [α _{x} α _{y} α _{z} 0 0 0]^{T} (input using, for example, ALPX, ALPY, ALPZ on MP command) |
ρ = density (input as DENS on MP command) |
C_{p} = specific heat at constant stress or pressure (input as C on MP command) |
Using {ε} and ΔT as independent variables, and replacing the entropy density S in Equation 10–31 by heat density Q using the second law of thermodynamics for a reversible change
(10–32) |
we obtain
(10–33) |
(10–34) |
where:
{β} = vector of thermoelastic coefficients = [D] {α} |
Substituting Q from Equation 10–34 into the heat flow equation Equation 6–1 produces:
(10–35) |
where:
K_{xx}, K_{yy}, K_{zz} = thermal conductivities (input as KXX, KYY, KZZ on MP command) |
Derivation of Thermoelastic Matrices
Applying the variational principle to stress equation of motion and the heat flow conservation equation coupled by the thermoelastic constitutive equations, produces the following finite element matrix equation:
(10–36) |
where:
[M] = element mass matrix (defined by Equation 2–58) |
[C] = element structural damping matrix (discussed in Damping Matrices) |
[K] = element stiffness matrix (defined by Equation 2–58) |
{u} = displacement vector |
{F} = sum of the element nodal force (defined by Equation 2–56) and element pressure (defined by Equation 2–58) vectors |
[C^{t}] = element specific heat matrix (defined by Equation 6–22) |
[K^{t}] = element thermal conductivity matrix (defined by Equation 6–22) |
{T} = temperature vector |
{Q} = sum of the element heat generation load and element convection surface heat flow vectors (defined by Equation 6–22) |
[B] = strain-displacement matrix (see Equation 2–44) |
{N} = element shape functions |
[C^{tu}] = element thermoelastic damping matrix = -T_{0}[K^{ut}]^{T} |
Energy Calculation
In static and transient thermoelastic analyses, the element instantaneous total strain energy is calculated as:
(10–37) |
where:
U_{t} = total strain energy (output as an NMISC element item UT). |
Note that Equation 10–37 uses the total strain, whereas the standard strain energy (output as SENE) uses the elastic strain.
In a harmonic thermoelastic analysis, the time-averaged element total strain energy is given by:
(10–38) |
where:
{ε}* = complex conjugate of the total strain |
The real part of Equation 10–38 represents the average stored strain energy, while its imaginary part - the average energy loss due to thermoelastic damping.
The thermoelastic damping can be quantified by the quality factor Q derived from the total strain energy Equation 10–38 using the real and imaginary solution sets:
(10–39) |
where:
N_{e} = number of thermoelastic elements |