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=Problem 4.2: Heat Conduction Fish & Belytschko Page 73 Problem 3.5 =

Given: The Following Boundary Conditions
Where (4.2.1) is the essential boundary condition defined at $$ \displaystyle x = 0 $$ and (4.2.2) is the natural boundary condition defined at $$ \displaystyle x = 10 $$ on the domain $$ \Omega = ]0,10[ \quad $$

Find:The Weak Form for 1-D Heat Conduction From the Strong Form
Obtain the weak form for the equation of the heat conduction with the boundary conditions stated above. The condition on the natural boundary condition is a convection condition.

Solution
The strong form has the following equation for a 1-D heat conduction body.

Assuming a steady-state from for the heat conduction and a constant area ( 4.2.3) reduces to

Multiplying by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.2.5) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.2.2) $$ k \frac{\partial T}{\partial x}_{x=10} = hT $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle T(x) $$ such that for all smooth $$ \displaystyle T(x) $$ with $$ \displaystyle T(0) = 100 $$ (4.2.8) is satisfied.

=Problem 4.4: Torsion of a Bar Fish & Belytschko Page 73 Problem 3.7 =

Given
Given the strong form for the circular bar in torsion (Fig. 1):

Where the natural boundary condition is

Where the essential boundary condition is



Where the essential boundary condition $$ \displaystyle m(x) $$ is a distributed moment per unit length, M is the torsion moment, $$ \displaystyle \phi $$ is the angle of rotation, $$ \displaystyle G $$ is the shear modulus and $$ \displaystyle J $$ is the polar moment of intertia given by $$ \displaystyle J = \frac{\pi C^4}{2} $$, where $$ \displaystyle C $$ is the radius of the circular shaft.

Find
a. Construct the weak form for the circular bar in torsion.

b. Assume that $$ \displaystyle m(x) = 0 $$ and integrate the differential equation given above. Find the integration constants using the boundary conditions.

(a)
Multiplying (4.4.1) by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.4.4) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.4.2) $$ JG \frac{\partial \phi}{\partial x}_{x=l} = M $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle \phi(x) $$ such that for all smooth $$ \displaystyle \phi(x) $$ with $$ \displaystyle \phi(0) = 0 $$ (4.4.7) is satisfied.

(b)
Applying equation (4.4.1) and with $$ \displaystyle m(x) = 0 $$ leads to the following form.

Integrating (4.4.8) with respect to x generates a constant of integration and provides the following form.

Applying the natural boundary condition from (4.4.2) leads to the following

Solving (4.4.9) for $$ \displaystyle \frac{\partial \phi}{\partial x} $$ and substituting in $$ \displaystyle C_1 = -M $$ leads to the following

Integrating (4.4.11) with respect to x generates a constant of integration and provides the following form, which is the well known equation for angle of twist of a bar.

Applying the essential boundary condition from (4.4.3) to (4.4.12) leads to the following

Therefore the constants and the angle of twist are as follows.

=Problem 4.5: =