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Inaccuracies in the formulas of the Materials Browser

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Let's work with Structural Mechanics Module> Thermal stress.

The temperature at which there is no change in size of your model, called Tref.

L (T) - the size of your model at a temperature T.

To solve the problem of thermal stress, we need the coefficient of thermal expansion

alfa_theor (T):

(L(T)-L(Tref))/L(Tref)=alfa_theor(T)*(T-Tref)

In the Model Library for the desired material we have coefficient alfa_solid (T), which is the coefficient of thermal expansion for Tref = 293 K:

(L(T)-L(293))/L(293)= alfa_solid(T)*(T-293)

Therefore, for some other from 293 K Tref, recorded

alfa(T)=alpha_solid(T)+(Tref-293)/(T-Tref)*(alpha_solid(T)-alpha_solid(Tref))

Multiplying this alfa(T) to (T-Tref):

alfa(T) *(T-Tref)= alpha_solid(T)(T-293)- alpha_solid(Tref)(Tref-293)=(L(T)-L(Tref))/L(293),

and this is not the same value, which we would like.

Correct value is:

alfa_theor(T)=(alpha_solid(T)+(Tref-293)/(T-Tref)*(alpha_solid(T)-alpha_solid(Tref)))/(1+ alpha_solid(Tref)(Tref-293))

or in Comsol description

(alpha_solid_1(T[1/K])[1/K]+(Tempref-293[K])/(T-Tempref)*(alpha_solid_1(T[1/K])[1/K]-alpha_solid_1(Tempref[1/K])[1/K]))/(1+ alpha_solid_1(Tempref[1/K])[1/K]*(Tempref-293))

The same goes for parameter dL:

dL= (L(T)-L(Tref))/L(Tref)

In the Materials Browser for the desired material we have coefficient

dL_solid(T)= (L(T)-L(293))/L(293)

Hence L(T)=L(293)(1+dL_solid(T)) and

dL=(dL_solid(T)-dL_solid(Tref))/(1+dL_solid(Tref))

Is that correct?

1 Reply Last Post 15.02.2013, 11:42 GMT-5
Henrik Sönnerlind COMSOL Employee

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Posted: 1 decade ago 15.02.2013, 11:42 GMT-5
Hi Ivan,

Your analysis is correct.

The factor (1+ alpha_solid(Tref)(Tref-293)) is however very small under all normal circumstances, taking into account possible actual values of temperatures and coefficients of thermal expansion. So this correction is currently ignored. Even with an extremely high reference temperature, it will hardly exceed 1.01 and for more common cases it would be much closer to unity.

Ignoring this factor essentially means that the initial "reference" length of the object is taken to be the same at the reference temperature as at 293 K.

Still, since the expression is already complicated, there is no need to ignore the correction term, and we will include it in the next version of COMSOL.

For those interested in how these expressions are derived:

Let be the reference temperature for the measured data (here 293 K). By definition



When we have a strain free temperature ('reference temperature') , which may not be the same temperature as the one on which the material data is based, then the same type of relation is expected to hold



Now is the 'strain free based' coefficient of thermal expansion.



Using (1) with :



Inserting (1) and (3) into (2) we will arrive at the expression you presented:



Regards,
Henrik


Hi Ivan, Your analysis is correct. The factor (1+ alpha_solid(Tref)(Tref-293)) is however very small under all normal circumstances, taking into account possible actual values of temperatures and coefficients of thermal expansion. So this correction is currently ignored. Even with an extremely high reference temperature, it will hardly exceed 1.01 and for more common cases it would be much closer to unity. Ignoring this factor essentially means that the initial "reference" length of the object is taken to be the same at the reference temperature as at 293 K. Still, since the expression is already complicated, there is no need to ignore the correction term, and we will include it in the next version of COMSOL. For those interested in how these expressions are derived: Let [math]T_m[/math] be the reference temperature for the measured data (here 293 K). By definition [math]L(T)=(1+\alpha_m(T) (T-T_m)) L(T_m) \quad(1)[/math] When we have a strain free temperature [math]T_{ref}[/math] ('reference temperature') , which may not be the same temperature as the one on which the material data is based, then the same type of relation is expected to hold [math]L(T)=(1+\alpha_r(T) (T-T_{ref}) )L(T_{ref})[/math] Now [math]\alpha_r[/math] is the 'strain free based' coefficient of thermal expansion. [math]\alpha_r(T) = \frac{L(T)-L(T_{ref})}{(T-T_{ref}) L(T_{ref})} \quad(2) [/math] Using (1) with [math]T = T_{ref}[/math]: [math]L(T_{ref})=(1+\alpha_m(T_{ref}) (T_{ref}-T_m)) L(T_m) \quad(3)[/math] Inserting (1) and (3) into (2) we will arrive at the expression you presented: [math]\alpha_r(T) =\Big( \alpha_m(T)+ \frac{(T_{ref}-T_m)(\alpha_m(T)-\alpha_m(T_{ref}))}{T-T_{ref}} \Big)\Big(1+\alpha_m(T_{ref})(T_{ref}-T_m)\Big)^{-1} [/math] Regards, Henrik

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