Examples:
- Principle of Virtual Work.
- Principle of Minimum Potential Energy.
- Principle of Minimum Complementary Energy.
- Hu-Washizu Variational Principle.
- Hellinger-Reissner Variational Principle.
Why ?
- Powerful way of approaching problems in linear elasticity.
- Can be used to derive the governing equations and boundary conditions for special classes of problems.
- Used as the basis of approximate solutions of elasticity problem, e.g., finite element method.
- Can be used to obtain rigorous bounds on the stiffness of elastic structures/solids.
A functional is basically a function of some other functions.
Let
be the displacement. Then the local strain energy density
is a functional.
Find
such that
![{\displaystyle U[u(x)]=\int _{x_{0}}^{x_{1}}F(x,u,u')dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f67a44ac97b0bfb0a7d6fcdaac827405c1231b1)
is a minimum.
Suppose
![{\displaystyle U[u(x)+\delta u(x)]\geq U[u(x)]~\forall ~|\delta u(x)|<h~{\text{and}}~x\in (x_{0},x_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/435894fb4468f89814c88585bab91fc8c124b157)
and equality holds only when
. Then
is the
variation of
.
A necessary condition that
minimizes
is that the
Euler equation
![{\displaystyle {\frac {\partial F}{\partial u}}-{\frac {d}{dx}}\left({\frac {\partial F}{\partial u^{'}}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fd4a27a0348e35526f7834afc3620c71b6c137)
is satisfied and
![{\displaystyle \left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{0}}=0~~{\text{or,}}~~\eta (x_{0})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd968b853c5aca3761ff74c7317baa0412258936)
and,
![{\displaystyle \left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{1}}=0~~{\text{or,}}~~\eta (x_{1})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc60cfe05caa954d80b527250cb5ba1506e8f17)
where
![{\displaystyle \delta u(x)=\epsilon \eta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2edf36d9d41f43309dd7bc782ea6327ef11542)
and
is small and
is arbitrary.
The imposed BCs are the conditions
![{\displaystyle \eta (x_{0})=0~{\text{and}}~\eta (x_{1})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81c7809ffeb9be7a61950bbc85f045bad977c34a)
These are automatically satisfied.
The natural BCs are the conditions
![{\displaystyle \left.{\frac {\partial }{\partial u'}}{F}\right|_{x_{0}}=0~{\text{and}}\left.{\frac {\partial }{\partial u'}}{F}\right|_{x_{1}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6743e56b714c29310eca675edc3229da774663db)
Any
that satisfies the necessary conditions make the functional
stationary and is said to be a stationary function
of the functional.
Suppose that
is a functional with
![{\displaystyle U[u(x)]=\int _{x_{0}}^{x_{1}}F(x,u,u')dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f67a44ac97b0bfb0a7d6fcdaac827405c1231b1)
Suppose that
is a small variation of
that satisfies
![{\displaystyle \left|\delta u(x)\right|\ll 1~;~~\left|\delta u'(x)\right|\ll 1~\forall ~x\in (x_{0},x_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b303a8dc5bb1c9a443068aa4aeea0e1a7591aad)
Then the variation of
is
![{\displaystyle \delta F=F(x,u+\delta u,u'+\delta u')-F(x,u,u')\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1a38500795fe4f89471a03c6fc7768f357fcfb)
or,
![{\displaystyle \delta F={\frac {\partial F}{\partial u}}~\delta u+{\frac {\partial F}{\partial u^{'}}}~\delta u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9a956b7f3904945d58f2dfe4d5e9ccf93d0456)
The variation of
is
![{\displaystyle \delta U=\int _{x_{0}}^{x_{1}}\delta Fdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5b0912ec71a4692b9bb30d5075b9556591f66a)
or,
![{\displaystyle \delta U=\left.{\frac {\partial F}{\partial u'}}~\delta u\right|_{x_{0}}^{x_{1}}+\int _{x_{0}}^{x_{1}}\left[{\frac {\partial F}{\partial u}}~\delta u-{\frac {d}{dx}}\left({\frac {\partial F}{\partial u'}}\right)\delta u\right]dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9329a9cbf05c9faf5b57ab2628d26eefcd3f3d3b)
Assuming that
is a necessary condition to minimize
, we get the same necessary conditions as before.
If there are additional constraints on minimization, we usually use
Lagrange Multipliers.
Suppose the additional constraint is that
.
Then, we define a function,
![{\displaystyle {\tilde {F}}(x,u,u',\lambda )=F(x,u,u')-\lambda \left[x+u+u'-C\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad379ca93cb501d66414dfda2857dc0a8374d3e3)
where
is the Lagrange multiplier.
Then,
![{\displaystyle {\frac {\partial {\tilde {F}}}{\partial \lambda }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3b8c68db592cd720c5a52f56c9b43be8dc16e2)
Then, the values that minimize the function subject to the
given constraint are given by the equations
![{\displaystyle {\frac {\partial {\tilde {F}}}{\partial \lambda }}=0~~;~~{\frac {\partial {\tilde {F}}}{\partial u}}=0~~;~~{\frac {\partial {\tilde {F}}}{\partial u'}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c68b8900d714e8a8a1761fe927f6f4ce25c962b)
Recall that the strain energy density is defined as
![{\displaystyle U({\boldsymbol {\varepsilon }})=\int _{0}^{\varepsilon }{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e747900cf434e40a9f773861639b68fed1527ea)
If the strain energy density is path independent, then it acts
as a potential for stress, i.e.,
![{\displaystyle \sigma _{ij}={\frac {\partial U(\varepsilon )}{\partial \varepsilon _{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc5d6abbfedadce5c16b130897429431d3668a3)
For adiabatic processes,
is equal to the change in internal energy per unit volume.
For isothermal processes,
is equal to the Helmholtz free energy per unit volume.
The natural state of a body is defined as the state in which the body is in stable thermal equilibrium with no external loads and zero stress and strain.
When we apply energy methods in linear elasticity, we implicitly assume that a body returns to its natural state after loads are removed. This implies that the Gibbs condition is satisfied :
![{\displaystyle U({\boldsymbol {\varepsilon }})\geq 0~~{\text{with}}~~U({\boldsymbol {\varepsilon }})=0~~{\text{iff}}~~{\boldsymbol {\varepsilon }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44cbc9d200251759718070be442c3f2065acca35)
This principle is used in the derivation of several minimization principles
and states that:
If
is a state of stress satisfying equilibrium
![{\displaystyle {\boldsymbol {\nabla }}\bullet {{\boldsymbol {\sigma }}^{(1)}}+\mathbf {f} ^{(1)}=0~~{\text{on}}~B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026cb608defae79a44890d39ea48aca38ddd8d22)
and the traction boundary condition
![{\displaystyle {\widehat {\mathbf {n} }}{}\bullet {\boldsymbol {\sigma }}^{(1)}=\mathbf {t} ^{(1)}~~{\text{on}}~\partial B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a37b416b6aea6731ab345f81db8500a87673b3)
Also, if
is a displacement field on
such that the strain
field
is given by
![{\displaystyle \varepsilon ^{(2)}={\frac {1}{2}}\left({\boldsymbol {\nabla }}u^{(2)}+({\boldsymbol {\nabla }}u^{(2)})^{T}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b544bc1ed3cb58d66c45b979fc9c69b017c4ad5e)
then
![{\displaystyle \int _{B}f_{i}^{(1)}u_{i}^{(2)}~dV+\int _{\partial B}t_{i}^{(1)}u_{i}^{(2)}~dA=\int _{B}\sigma _{ij}^{(1)}\varepsilon _{ij}^{(2)}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ddcaac81ca276a7facb1a33026604cbd43a2603)
The converse also holds - and is usually more interesting because it
gives us a different way of thinking about equilibrium.
If there are jump discontinuities in a body, then what does equilibrium
imply ?
Suppose that
has a jump discontinuity across a body
along the surface
with normal
because the materials
on the two sides are different.
We define the equilibrium state to be one that satisfies the principle
of virtual work for all displacement fields.
Now, if the spin tensor is zero, then
![{\displaystyle \sigma _{ij}^{(1)}\varepsilon _{ij}^{(2)}=(\sigma _{ij}^{(1)}u_{i}^{(2)})_{,j}-\sigma _{ij,j}^{(1)}u_{i}^{(2)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a78ace7985a0d0e0ad1bf2778bfbeb0deb778123)
If we use the above, and apply the divergence theorem to the virtual
work equation we get
![{\displaystyle \int _{B}\left(\sigma _{ji,j}^{(1)}+f_{i}^{(1)}\right)u_{i}^{(2)}~dV+\int _{\partial B}\left(t_{i}^{(1)}-\sigma _{ij}^{(1)}n_{j}\right)u_{i}^{(2)}~dA=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fae16ff82ae79e5fdc54408464d14791459a4f5)
For the stress jump to satisfy this equation, we must have
![{\displaystyle \int _{B}\left(\sigma _{ji,j}^{(1)}+f_{i}^{(1)}\right)u_{i}^{(2)}~dV+\int _{\partial B}\left(t_{i}^{(1)}-\sigma _{ij}^{(1)}n_{j}\right)u_{i}^{(2)}~dA+\int _{S}(\sigma _{ij}^{+}m_{j}^{+}-\sigma _{ij}^{-}m_{j}^{-})u_{i}^{(2)}dS=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5dc9e5cc094d83198cad79957cbd9f70db6cdee)
Hence, equilibrium is satisfied when
![{\displaystyle \sigma _{ij}^{+}m_{j}^{+}-\sigma _{ij}^{-}m_{j}^{-}=0~~{\text{on}}~~S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3843c39b6cb2b7b805b140235f6a19fe119a2911)
which means that even though a jump can exist in the stresses, the
tractions have to be continuous across the discontinuity.
The work done by external forces on a body
can be represented
as a functional
![{\displaystyle W[\mathbf {u} ]=\int _{\partial B}\left(\int _{0}^{\mathbf {u} }\mathbf {t} \bullet d\mathbf {u} \right)dA+\int _{B}\left(\int _{0}^{\mathbf {u} }\mathbf {f} \bullet d\mathbf {u} \right)dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1714b19e42e2fd164a9b0fc3190448dcd4d986)
Taking the variation of
, we get
![{\displaystyle \delta W=\int _{\partial B}{\frac {\partial }{\partial \mathbf {u} }}\left[\left(\int _{0}^{\mathbf {u} }\mathbf {t} \bullet d\mathbf {u} \right)\right]dA+\int _{B}{\frac {\partial }{\partial \mathbf {u} }}\left[\left(\int _{0}^{\mathbf {u} }\mathbf {f} \bullet d\mathbf {u} \right)\right]dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9f667fef4b8bd3137aa37ab89584dc11bc54a1)
In index notation,
![{\displaystyle \delta W=\int _{\partial B}{\frac {\partial }{\partial u_{j}}}\left[\left(\int _{0}^{\mathbf {u} }t_{i}~du_{i}\right)\right]\delta u_{j}~dA+\int _{B}{\frac {\partial }{\partial u_{j}}}\left[\left(\int _{0}^{\mathbf {u} }f_{i}~du_{i}\right)\right]\delta u_{j}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31bd09b534e21b51d614c092fb1427c71980b9d2)
Noting that the external forces and body forces are not functions
of
, the above equation reduces to
![{\displaystyle \delta W=\int _{\partial B}t_{j}~\delta u_{j}~dA+\int _{B}f_{j}~\delta u_{j}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1120c8d8ded1cb35398dc7e4793bb538eb774592)
The above expression is called the external virtual work.
If we apply the principle of virtual work, with
![{\displaystyle \mathbf {t} =\mathbf {t} ^{(1)}~;~~\mathbf {f} =\mathbf {f} ^{(1)}~;~~{\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{(1)}~;~~\delta \mathbf {u} =\mathbf {u} ^{(2)}~;~~\delta {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}^{(2)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb66831ca71199b3cdc2e32ff74a4219c88b679f)
we get
![{\displaystyle \delta W=\int _{B}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6df1b68b89ba6b6a34cdeea4f4e4b91abdf718cf)
or,
![{\displaystyle \delta W=\int _{B}{\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{ij}}}\delta \varepsilon _{ij}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9c95887b341f7db0fab9cf1d07b5fc634f392e)
This is the expression for the internal virtual work.
Thus, another form of the principle of virtual work is
![{\displaystyle \int _{B}{\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{ij}}}\delta \varepsilon _{ij}~dV=\int _{\partial B}t_{j}~\delta u_{j}~dA+\int _{B}f_{j}~\delta u_{j}~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7cbd2448bd3239f59b0d3ad3935f4d534e4e35)
Doing the reverse operation, it can be shown that
![{\displaystyle W[{\boldsymbol {\varepsilon }}(\mathbf {x} )]=\int _{R}U({\boldsymbol {\varepsilon }})~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a49815915e11855ec47f6285527b305fde050aec)
which relates the strain energy density (
) to the functional
that represents the work done by external forces.
Developed and explored by Green (1839), Haughton(1849), Kirchhoff (1850),
Love (1906), Trefftz (1928) and others.
This principle states that:
- Among all possible kinematically admissible displacement fields
- the potential energy functional is rendered stationary
- by only those that are actual displacement fields.
This principle states that
- If the prescribed traction and body force fields are independent of the deformation
- then the actual displacement field makes the potential energy functional an absolute minimum.
Consider a body
with a boundary
with an applied
body force field
.
Suppose that
displacement BCs
are prescribed on the part of
the boundary
.
Suppose also that traction BCs
are applied on the portion of the
boundary
.
A displacement field
is kinematically admissible if
satisfies the displacement boundary conditions
on
.
is continuously differentiable, i.e.,
and
.
The potential energy functional associated with the
kinematically admissible displacement field
is defined as
![{\displaystyle \Pi [\mathbf {v} ,{\boldsymbol {\varepsilon }},{\boldsymbol {\sigma }}]=\int _{B}U({\boldsymbol {\varepsilon }})~dV-\int _{B}{\tilde {\mathbf {f} }}\bullet \mathbf {v} ~dV-\int _{\partial B^{t}}{\tilde {\mathbf {t} }}\bullet \mathbf {v} ~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b08b104b015bb5d194d70745117d8e2d8941edc)
or,
![{\displaystyle \Pi [\mathbf {v} ]={\frac {1}{2}}\int _{B}{\boldsymbol {\nabla }}{\mathbf {v} }:\mathbf {C} :{\boldsymbol {\nabla }}{\mathbf {v} }~dV-\int _{B}{\tilde {\mathbf {f} }}\bullet \mathbf {v} ~dV-\int _{\partial B^{t}}{\tilde {\mathbf {t} }}\bullet \mathbf {v} ~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a774ec7f954ac37d9d67ddb5507e8cecdc6ea4)
In index notation,
![{\displaystyle \Pi [\mathbf {v} ]={\frac {1}{2}}\int _{B}C_{ijkl}~v_{k,l}~v_{i,j}~dV-\int _{B}{\tilde {f}}_{i}~v_{i}~dV-\int _{\partial B^{t}}{\tilde {t}}_{i}~v_{i}~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5f228c6cdbd97f1872150e4aa3a48de851c1c9)
Stationary Points and Minimum of the Potential Energy Functional
[edit | edit source]
What do we mean when we say that we "render the potential energy
functional stationary" or "minimum"? Note that the potential energy
is a functional of a vector field.
Suppose that the actual displacement field (one that satisfies equilibrium,
compatibility and the boundary conditions) is
.
Let
be a kinematically admissible variation of
, i.e.,
![{\displaystyle \mathbf {v} =\mathbf {u} +k~\delta \mathbf {u} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a822bf3456586101a6614a6fb1e3a13ee16134)
where
is a constant.
Then
must be a displacement field that is continuously
differentiable and satisfies the boundary conditions
![{\displaystyle \delta \mathbf {u} =0~~{\text{on}}~~\partial B^{u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2902ef593e24f22a66e8224c5056e01fa3214e9c)
The potential energy functional for
is
![{\displaystyle {\begin{aligned}\Pi [\mathbf {u} +k~\delta \mathbf {u} ]=&{\frac {1}{2}}\int _{B}{\boldsymbol {\nabla }}{(\mathbf {u} +k~\delta \mathbf {u} )}:\mathbf {C} :{\boldsymbol {\nabla }}{(\mathbf {u} +k~\delta \mathbf {u} )}~dV\\&-\int _{B}{\tilde {\mathbf {f} }}\bullet (\mathbf {u} +k~\delta \mathbf {u} )~dV-\int _{\partial B^{t}}{\tilde {\mathbf {t} }}\bullet (\mathbf {u} +k~\delta \mathbf {u} )~dA\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2996866217e8f28604e11ac4bb4cf4728f5969a1)
In index notation,
![{\displaystyle {\begin{aligned}\Pi [\mathbf {u} +k~\delta \mathbf {u} ]=&{\frac {1}{2}}\int _{B}C_{ijkl}(u_{i,j}+k~\delta u_{i,j})(u_{k,l}+k~\delta u_{k,l})~dV\\&-\int _{B}{\tilde {f}}_{i}(u_{i}+k~\delta u_{i})~dV-\int _{\partial B^{t}}{\tilde {t}}_{i}(u_{i}+k~\delta u_{i})~dA\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0024baa86078dc66f071ab56216260c2ecdf9d93)
Expanding and rearranging,
![{\displaystyle {\begin{aligned}\Pi [\mathbf {u} +k~\delta \mathbf {u} ]=&{\frac {1}{2}}\int _{B}C_{ijkl}~u_{i,j}~u_{k,l}~dV-\int _{B}{\tilde {f}}_{i}~u_{i}~dV-\int _{\partial B^{t}}{\tilde {t}}_{i}~u_{i}~dA\\&+k\left[{\frac {1}{2}}\int _{B}C_{ijkl}~u_{i,j}~\delta u_{k,l}~dV+{\frac {1}{2}}\int _{B}C_{ijkl}~\delta u_{i,j}~u_{k,l}~dV\right]{\text{(1)}}\qquad \\&-k\left[\int _{B}{\tilde {f}}_{i}~\delta u_{i}~dV+\int _{\partial B^{t}}{\tilde {t}}_{i}~\delta u_{i}~dA\right]\\&+{\frac {k^{2}}{2}}\int _{B}C_{ijkl}~\delta u_{i,j}~\delta u_{k,l}~dV\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d289f7af6b5c254bec26f109ce593dde07ae166)
Using the symmetry of the stiffness tensor, we can simplify the above
expression and write it it terms of variations of
. Thus,
![{\displaystyle \Pi [\mathbf {u} +k~\delta \mathbf {u} ]=\Pi [\mathbf {u} ]+k~\delta \Pi [\mathbf {u} ,\delta \mathbf {u} ]+{\frac {1}{2}}k^{2}~\delta ^{2}\Pi (\mathbf {u} ,\delta \mathbf {u} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81f32dc0f0978f23d754dc3e841c6b289f7c3aa)
You can check that the first and second variations of
turn out to
be equal to the expanded terms in equation (1).
If
![{\displaystyle \delta \Pi (\mathbf {u} ,\delta \mathbf {u} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1ec3e669740c1526a4b5a1d2d024c3df43cf3d)
for all admissible variations
, then
is a { stationary
point} of the functional
.
If
![{\displaystyle \delta ^{2}\Pi (\mathbf {u} ,\delta \mathbf {u} )>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c51719a82376ffbea4f683fdcaaf9c757f4e036)
for all admissible variations
, then
is makes the
functional
a minimum.
- The potential energy functional is a global minimum if and only if the displacement field satisfies traction BCs, equilibrium and the displacement BCs.
- Thus, if the potential energy functional actually has a global mininum, then a solution exists and must be unique.
- Displacement boundary value problems do not face the problem of rigid body motions. Therefore, a global minimum always exists for such problems and is unique.
- Traction boundary value problems may not have unique solutions, nor might solutions always exist unless the external loads are in static equilibrium.
Suppose that we have a cylindrical body of length
and an arbitrary cross-section that is subject to equal and opposite torques at the two ends. The displacement field is given by
![{\displaystyle u_{1}=-\alpha x_{2}x_{3}~;~~u_{2}=\alpha x_{1}x_{3}~;~~u_{3}=\alpha \psi (x_{1},x_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/699580dda004e75edc7e7d7d310b533b43fb4d0e)
The traction-free boundary conditions on the lateral surfaces can be
given as
![{\displaystyle {\widehat {\mathbf {n} }}{}\bullet {\boldsymbol {\sigma }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ad22268488de5a405ee4f3f4ec1afe5fc8f28c)
The torque BC at the ends can be replaced with displacement BCs
![{\displaystyle {\text{on}}~x_{3}=0~;~~~~~~u_{1}=u_{2}=0~,~~u_{3}=\alpha \psi (x_{1},x_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba685109fad86a8b276cb4592ed2fdadd779a48)
and
![{\displaystyle {\text{on}}~x_{3}=L~;~~~~~~u_{1}=-\alpha Lx_{2}~,~~u_{2}=\alpha Lx_{1}~,~~u_{3}=\alpha \psi (x_{1},x_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194d12000f95cc0fbcf8032789ba96265c1b2af5)
Thus, we change the problem from a purely traction boundary value
problem to one in which the twist per unit length (
) is prescribed instead of the applied torque (
).
The modified problem is one with zero body force and zero tractions.
Therefore, the potential energy functional reduces to
![{\displaystyle \Pi [\mathbf {v} ,{\boldsymbol {\varepsilon }},{\boldsymbol {\sigma }}]=\int _{B}U({\boldsymbol {\varepsilon }})~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c33d7a52c3d15eb03bdc66b7e5272992b0771331)
The stresses and strains for the torsion problem are given by
![{\displaystyle {\begin{aligned}\sigma _{13}=\mu \alpha (\psi _{,1}-x_{2})&&\sigma _{23}=\mu \alpha (\psi _{,2}+x_{1})\\\varepsilon _{13}={\frac {\alpha }{2}}(\psi _{,1}-x_{2})&&\varepsilon _{23}={\frac {\alpha }{2}}(\psi _{,2}+x_{1})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0e33bb9c8314385171beba1f3251641ab4dc3e)
Therefore, the internal energy is
![{\displaystyle U({\boldsymbol {\varepsilon }})={\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}={\frac {1}{2}}\mu \alpha ^{2}\left[(\psi _{,1}-x_{2})^{2}+(\psi _{,2}+x_{1})^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7323c7738c55268f107b45f93a49fe51bb174fc8)
The potential energy per unit length (
) is
![{\displaystyle {\bar {\Pi }}[\psi (x_{1},x_{2})]={\frac {1}{2}}\mu \alpha ^{2}\int _{\mathcal {S}}\left[(\psi _{,1}-x_{2})^{2}+(\psi _{,2}+x_{1})^{2}\right]dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/809aab8affb4eb992705f3a3f9873493cffb8c76)
According to the principle of minimum potential energy, the actual
warping function is the one that makes
an absolute minimum.
Suppose we are given a warping function of the form
![{\displaystyle \psi =Ax_{1}^{2}+Bx_{1}x_{2}+Cx_{2}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4348123990ee9bf36009a9cab8b0cd572771c361)
Then, the potential energy per unit length is
![{\displaystyle {\begin{aligned}{\bar {\Pi }}[\psi (x_{1},x_{2})]={\frac {1}{2}}\mu \alpha ^{2}&\left[\int _{\mathcal {S}}\left[4A^{2}+(B+1)^{2}\right]x_{1}^{2}~dA+\int _{\mathcal {S}}\left[4C^{2}+(B-1)^{2}\right]x_{2}^{2}~dA+\right.\\&\left.\int _{\mathcal {S}}4\left[A(B-1)+C(B+1)\right]x_{1}~x_{2}~dA\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abbace2ec8bac3779ec23ef19ebecd378aa18bbb)
If
and
are the principal axes of inertia, then we have
![{\displaystyle I_{1}=\int _{\mathcal {S}}x_{1}^{2}~dA~;~~I_{2}=\int _{\mathcal {S}}x_{2}^{2}~dA~;~~I_{12}=\int _{\mathcal {S}}x_{1}~x_{2}~dA=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e565505789c0b753760bd7d0773f9bdf5e1413b)
Hence,
![{\displaystyle {\begin{aligned}{\bar {\Pi }}[\psi (x_{1},x_{2})]=&{\frac {1}{2}}\mu \alpha ^{2}\left[\left[4A^{2}+(B+1)^{2}\right]I_{1}+\left[4C^{2}+(B-1)^{2}\right]I_{2}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e70fd4f92491adde57d06c9fa00780ed3a92f7)
The stationary points of the potential energy functional are given by
![{\displaystyle {\begin{aligned}{\frac {\partial {\bar {\Pi }}}{\partial A}}&=4\mu \alpha ^{2}~I_{2}~A=0\\{\frac {\partial {\bar {\Pi }}}{\partial B}}&=\mu \alpha ^{2}\left[(B+1)I_{2}+(B-1)I_{1}\right]=0\\{\frac {\partial {\bar {\Pi }}}{\partial C}}&=4\mu \alpha ^{2}~I_{2}~A=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10757bdcabe92a851723ed9886e78c47d7d01439)
Thus, we have,
![{\displaystyle A=0~;~~B={\frac {I_{1}-I_{2}}{I_{1}+I_{2}}}~;~~C=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e633d342265df6bfdc1789fe1085d0172228cb60)
Thus the best approximation to the warping function is
![{\displaystyle \psi ={\frac {I_{1}-I_{2}}{I_{1}+I_{2}}}x_{1}x_{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e61ec9622dad20aded1e10be1cc1a731c6ea909)
The above technique is called the Rayleigh-Ritz method.
An important observation that should be made at this stage is about the approximate nature of the solution.
For cross-sections in which
, (e.g., circular or square sections) the best approximation for
is
. This gives us the exact result for circular cross-sections.
However, for square cross-sections we have an error of nearly 20%.
Introduction to Elasticity