User:Tmhoang81/Elastic stability

Principles of Mechanics

Principle of virtual work

In Lurie 2005, p.58, it is written that "The elementary work done by all external and internal forces due to virtual displacement of the continuum particles from their equilibirum position is equal to zero." The elementary work of the external forces, in Lurie 2005, p.130, is expressed in the form:

Remember to give a lot of references with page numbers immediately as you write so that you don't have to go back to add references, since you would forget where the page numbers were after a while. So as soon as you write about a new concept, or a new equation, immediately give the references. See the article Gradient of vector: Two tensor conventions for examples of how references with page numbers. Egm6321.f11 14:40, 17 June 2011 (UTC)

$\displaystyle {\begin{array}{l}\delta {A_{e}}=\int \limits _{S_{C}}{{\mathbf {t} }\bullet \delta {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}=\int \limits _{S_{C}}{{\boldsymbol {\sigma }}\bullet {\mathbf {n} }\bullet \delta {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}\\=\int \limits _{V_{C}}{{\boldsymbol {\nabla }}\bullet \left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {u} }}\right)dV}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}=\int \limits _{V_{C}}{\left({{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}}\right)\bullet \delta {\mathbf {u} }dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)dV}+\\+\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}=\int \limits _{V_{C}}{\underbrace {\left({{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}\,\,{\mathbf {+} }\,\rho {\mathbf {b} }}\right)} _{\mathbf {0} }\bullet \delta {\mathbf {u} }dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)dV}\end{array}}$

(1.1.1)

You can define directly here the virtual internal work $\delta A_{i}:=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)dV}\equiv \delta U$, instead of defining $\delta A_{i}:=-\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)dV}$ in Eq.(1.1.8) and Eq.(1.1.9). This definition is consistent with the definition of $\delta A_{i}$ in a comment box in section on The Lagrange's variational equation below. Eml5526.s11 14:48, 10 June 2011 (UTC)

Remember that the gradient operator acts in the spatial frame, so

$\displaystyle \nabla \delta {\mathbf {u} }={\frac {\partial \delta {\mathbf {u} }}{\partial {\mathbf {x} }}}\neq \delta {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}$

$\displaystyle {\nabla _{\mathbf {X} }}\delta {\mathbf {u} }={\frac {\partial \delta {\mathbf {u} }}{\partial {\mathbf {X} }}}=\delta {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}$

Refer to Gradient of vector: Two tensor conventions, and mention which convention used. Eml5526.s11 14:13, 3 June 2011 (UTC)

Decompose

$\displaystyle {\boldsymbol {\nabla }}\delta {\mathbf {u} }={\frac {1}{2}}\left[{{\boldsymbol {\nabla }}\delta {\mathbf {u} }+{{\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)}^{T}}}\right]+{\frac {1}{2}}\left[{{\boldsymbol {\nabla }}\delta {\mathbf {u} }\,-{{\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)}^{T}}}\right]=\delta {\mathbf {D} }+\delta {\mathbf {W} }$

(1.1.2)

then

$\displaystyle \delta {A_{e}}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {D} }}\right)dV}+\int \limits _{V_{C}}{\underbrace {tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {W} }}\right)} _{0}dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {D} }}\right)dV}$

(1.1.3)

It is obvious that in p.45

$\displaystyle \delta {\mathbf {F} }=\delta {\frac {\partial {\mathbf {x} }}{\partial {\mathbf {X} }}}=\delta {\frac {\partial \left({{\mathbf {u} }+{\mathbf {X} }}\right)}{\partial {\mathbf {X} }}}=\delta \left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{\mathbf {I} }}\right)=\delta {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}={\frac {\partial \delta {\mathbf {u} }}{\partial {\mathbf {X} }}}={\frac {\partial \delta {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\frac {\partial {\mathbf {x} }}{\partial {\mathbf {X} }}}={\boldsymbol {\nabla }}\delta {\mathbf {u} }\bullet {\mathbf {F} }$

$\displaystyle {\begin{array}{l}{\left({\delta {\mathbf {F} }}\right)^{T}}=\delta {{\mathbf {F} }^{T}}={{\mathbf {F} }^{T}}\bullet {\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)^{T}}\end{array}}$

(1.1.4)

So

$\displaystyle {\begin{array}{l}\delta {\mathbf {C} }=\delta \left({{{\mathbf {F} }^{T}}\bullet {\mathbf {F} }}\right)=\delta {{\mathbf {F} }^{T}}\bullet {\mathbf {F} }+{{\mathbf {F} }^{T}}\bullet \delta {\mathbf {F} }={{\mathbf {F} }^{T}}\bullet {\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)^{T}}\bullet {\mathbf {F} }+{{\mathbf {F} }^{T}}\bullet {\boldsymbol {\nabla }}\delta {\mathbf {u} }\bullet {\mathbf {F} }\\={{\mathbf {F} }^{T}}\bullet \left[{{{\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)}^{T}}+{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right]\bullet {\mathbf {F} }=2{{\mathbf {F} }^{T}}\bullet \delta {\mathbf {D} }\bullet {\mathbf {F} }\\\end{array}}$

(1.1.5)

but

$\displaystyle {\mathbf {C} }={\mathbf {I} }+2{\mathbf {E} }\to \delta {\mathbf {C} }=2\delta {\mathbf {E=} }2{{\mathbf {F} }^{T}}\bullet \delta {\mathbf {D} }\bullet {\mathbf {F} }\leftrightarrow \delta {\mathbf {E=} }{{\mathbf {F} }^{T}}\bullet \delta {\mathbf {D} }\bullet {\mathbf {F} }\leftrightarrow \delta {\mathbf {D=} }{{\mathbf {F} }^{-T}}\bullet \delta {\mathbf {E} }\bullet {{\mathbf {F} }^{-1}}$

(1.1.6)

So the elementary work of the external forces (work done by surface forces and body forces) is:

$\displaystyle {\begin{array}{l}\delta {A_{e}}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {D} }}\right)dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {{\mathbf {F} }^{-T}}\bullet \delta {\mathbf {E} }\bullet {{\mathbf {F} }^{-1}}}\right)dV}\\=\int \limits _{V_{0}}{tr\left({{{\mathbf {F} }^{-1}}\bullet {\boldsymbol {\sigma }}\bullet {{\mathbf {F} }^{-T}}\bullet \delta {\mathbf {E} }}\right)\left({Jd{V_{0}}}\right)}=\int \limits _{V_{0}}{tr\left({\underbrace {J{{\mathbf {F} }^{-1}}\bullet {\boldsymbol {\sigma }}\bullet {{\mathbf {F} }^{-T}}} _{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)d{V_{0}}}=\int \limits _{V_{0}}{tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)d{V_{0}}}\\\end{array}}$

(1.1.7)

From the virtual principle work, the elementary work of the internal forces (work done by elastic forces) is:

$\displaystyle \delta {A_{i}}=-\delta {A_{e}}=-\int \limits _{V_{0}}{tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)d{V_{0}}}$

(1.1.8)

Now assume that internal elastic forces have a potential, $\displaystyle \delta {A_{i}}=-\delta U$ with

$\displaystyle \delta {A_{i}}=-\delta {A_{e}}=-\int \limits _{V_{0}}{tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)d{V_{0}}}$

(1.1.9)

where $\displaystyle u$ is internal energy per unit mass (stored elastic energy)

$\displaystyle \to \delta U=\delta \int \limits _{V_{C}}{\rho udV}=\delta \int \limits _{V_{0}}{{\rho _{0}}ud{V_{0}}=}\int \limits _{V_{0}}{\delta \left({{\rho _{0}}u}\right)d{V_{0}}}$

(1.1.10)

From this we refer:

$\displaystyle \delta U=\int \limits _{V_{0}}{\delta \left({{\rho _{0}}u}\right)d{V_{0}}}=\int \limits _{V_{0}}{\delta Wd{V_{0}}}=\int \limits _{V_{0}}{tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)d{V_{0}}}\leftrightarrow \delta W=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)$

(1.1.11)

where $\displaystyle W={\rho _{0}}u$ is the energy per unit volume. From the definition:

$\displaystyle \delta W=tr\left({{\frac {\partial W}{\partial {\mathbf {E} }}}\bullet \delta {\mathbf {E} }}\right)=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet \delta {\mathbf {E} }}\right)\to {\boldsymbol {\tilde {\sigma }}}={\frac {\partial W}{\partial {\mathbf {E} }}}$

(1.1.12)

If we now consider infinitesimal displacement gradients $\displaystyle W={\rho _{0}}u$ then

$\displaystyle {\boldsymbol {\tilde {\sigma }}}={\frac {\partial W}{\partial {\mathbf {E} }}}={\left.{\frac {\partial W}{\partial {\mathbf {E} }}}\right|_{{\mathbf {E} }={\mathbf {0} }}}+{\left.{\frac {{\partial ^{2}}W}{\partial {{\mathbf {E} }^{2}}}}\right|_{{\mathbf {E} }={\mathbf {0} }}}{\mathbf {E} }+{\mathbf {O} }\left({{\left|{\mathbf {E} }\right|}^{2}}\right)$

(1.1.13)

Assume the unstrained body possesses no residual stress and define fourth-rank elastic modulus tensor

$\displaystyle {\mathbf {C=} }{\left.{\frac {{\partial ^{2}}W}{\partial {{\mathbf {E} }^{2}}}}\right|_{{\mathbf {E} }={\mathbf {0} }}}\to {\boldsymbol {\tilde {\sigma }}}={\boldsymbol {\sigma }}={\frac {\partial W}{\partial {\mathbf {E} }}}={\mathbf {CE} }$

(1.1.14)

Having established the fobold of above linear relations between stresses and strains, we can integrate to get:

$\displaystyle W={\frac {1}{2}}{\mathbf {ECE=} }{\frac {1}{2}}tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {E} }}\right)={\frac {\lambda }{2}}tr{\left({\mathbf {E} }\right)^{2}}+\mu tr\left({{\mathbf {E} }^{2}}\right)={\frac {1}{4\mu }}\left[{tr\left({{\boldsymbol {\sigma }}^{2}}\right)-{\frac {\nu }{1+\nu }}tr{{\left({\boldsymbol {\sigma }}\right)}^{2}}}\right]$

(1.1.15)

The Lagrange’s variational equation

Consider a certain state of equilibrium or of small elastic vibrations of a body, which is charaterized by the stresses $\displaystyle {\boldsymbol {\sigma }}$ and the displacements $\displaystyle u$. From

$\displaystyle {\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}\,\,{\mathbf {+} }\,\rho {\mathbf {b=} }\rho {\frac {{\partial ^{2}}{\mathbf {u} }}{\partial {t^{2}}}}\to \int \limits _{V_{C}}{\left({{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}\,\,{\mathbf {+} }\,\rho {\mathbf {b} }-\rho {\frac {{\partial ^{2}}{\mathbf {u} }}{\partial {t^{2}}}}}\right)\bullet {{\mathbf {u} }^{'}}dV}=0$

(1.2.1)

Actually, the most general form of the Cauchy equation of motion that is also valid for fluids is written as follows: ${\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}\,\,{\mathbf {+} }\,\rho {\mathbf {b=} }\rho {\frac {{D}{\mathbf {v} }}{D{t}}}$, where ${\mathbf {v} }$ is the velocity field, and the acceleration is the material time derivative of the velocity, written as ${\frac {{D}{\mathbf {v} }}{D{t}}}$. See Eq.(38) in Kolmogorov scales.

Eml5526.s11 14:21, 3 June 2011 (UTC)

where $\displaystyle {{\mathbf {u} }^{'}}$ are arbitrary functions which have continuous partial derivatives of the first and second order with respect to spatial variables and time. Using divergence theorem to convert:

$\displaystyle {\begin{array}{l}\int \limits _{V_{C}}{\left({{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}}\right)\bullet {{\mathbf {u} }^{'}}dV}=\int \limits _{V_{C}}{\left[{{\boldsymbol {\nabla }}\bullet \left({{\boldsymbol {\sigma }}\bullet {{\mathbf {u} }^{'}}}\right)-{\boldsymbol {\sigma }}\cdot \cdot {\boldsymbol {\nabla }}{\mathbf {v} }}\right]dV=\int \limits _{S_{C}}{{\mathbf {n} }\bullet {\boldsymbol {\sigma }}\bullet {{\mathbf {u} }^{'}}dS}}-\int \limits _{V_{C}}{{\boldsymbol {\sigma }}\cdot \cdot {\boldsymbol {\nabla }}{\mathbf {v} }dV}\\=\int \limits _{S_{C}}{{\mathbf {t} }\bullet {{\mathbf {u} }^{'}}dS-\int \limits _{V_{C}}{{\boldsymbol {\sigma }}\cdot \cdot {\boldsymbol {\nabla }}{{\mathbf {u} }^{'}}dV}}=\int \limits _{S_{C}}{{\mathbf {t} }\bullet {{\mathbf {u} }^{'}}dS-\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {{\mathbf {e} }^{'}}}\right)dV}}\\\end{array}}$

(1.2.2)

Define the "horizontal" double contraction "$\cdot \cdot$", and give reference, e.g., Malvern 1969. Eml5526.s11 14:21, 3 June 2011 (UTC)

So,

$\displaystyle \int \limits _{S_{C}}{{\mathbf {t} }\bullet {{\mathbf {u} }^{'}}dS+\int \limits _{V_{C}}{\,\rho \left({{\mathbf {b} }-{\frac {{\partial ^{2}}{\mathbf {u} }}{\partial {t^{2}}}}}\right)\bullet {{\mathbf {u} }^{'}}dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {{\mathbf {e} }^{'}}}\right)dV}}$

(1.2.3)

The above equation is essentially $\delta A_{e}-\delta A_{m}=\delta A_{i}$, where the external virtual work is $\delta A_{e}:=\int \limits _{S_{C}}{{\mathbf {t} }\bullet {{\mathbf {u} }^{'}}dS+\int \limits _{V_{C}}{\,\rho \mathbf {b} \bullet {{\mathbf {u} }^{'}}dV}}$, the inertial virtual work $\delta A_{m}:=\int \limits _{V_{C}}{\,\rho {\frac {{\partial ^{2}}{\mathbf {u} }}{\partial {t^{2}}}}}\bullet {{\mathbf {u} }^{'}}dV$, and the internal virtual work $\delta A_{i}:=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {{\mathbf {e} }^{'}}}\right)dV}$.
Usually, in FEM, the above "weak form" is written as follows $\delta A_{m}+\delta A_{i}=\delta A_{e}$, which leads to the discrete weak form $c^{T}\left(M{\ddot {d}}+Kd\right)=c^{T}F,\ \forall c$ and then to semi-discrete equation $M{\ddot {d}}+Kd=F$.

Eml5526.s11 14:21, 3 June 2011 (UTC)

where

$\displaystyle {{\mathbf {e} }^{'}}={\frac {1}{2}}\left[{{\boldsymbol {\nabla }}{{\mathbf {u} }^{'}}+{{\left({{\boldsymbol {\nabla }}{{\mathbf {u} }^{'}}}\right)}^{T}}}\right]$

(1.2.4)

Now let the displacements $\displaystyle {{\mathbf {u} }^{'}}$ be the actual displacements $\displaystyle {\mathbf {u} }$ and assume that the state of stress corresponds to the equilibrium of the body, then the equality reduces:

$\displaystyle \int \limits _{S_{C}}{{\mathbf {t} }\bullet {\mathbf {u} }dS+\int \limits _{V_{C}}{\,\rho {\mathbf {b} }\bullet {\mathbf {u} }dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {e} }}\right)dV}=2\int \limits _{V_{C}}{WdV}}$

(1.2.5)

The inertial force disappeared here; there is no need to remove it. Mention that you are now dealing with static equilibrium, not dynamic equilibrium, so you ignore the inertial term. Also, ${\rm {tr}}({\boldsymbol {\sigma }}\cdot {\boldsymbol {e}})={\boldsymbol {\sigma }}:{\boldsymbol {e}}$. Eml5526.s11 14:21, 3 June 2011 (UTC)

Now let the displacements $\displaystyle {{\mathbf {u} }^{'}}$ be the virtual displacements $\displaystyle \delta \mathbf {u}$ and assume that the state of stress corresponds to the equilibrium of the body, then the equality reduces to principle of virtual work:

$\displaystyle \int \limits _{S_{C}}{{\mathbf {t} }\bullet \delta {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {e} }}\right)dV}$

(1.2.6)

where

$\displaystyle \delta {\mathbf {e} }={\frac {1}{2}}\left[{{\boldsymbol {\nabla }}\delta {\mathbf {u} }+{{\left({{\boldsymbol {\nabla }}\delta {\mathbf {u} }}\right)}^{T}}}\right]$

(1.2.7)

Remember that in deriving above formular, we fixed quantities $\displaystyle {\mathbf {t} },{\boldsymbol {\sigma }}$ and $\displaystyle \rho {\frac {{\partial ^{2}}{\mathbf {u} }}{\partial {t^{2}}}}$ while varying the displacements $\displaystyle {\mathbf {u} }$ and corresponding strains. So,

$\displaystyle {\begin{array}{l}\int \limits _{S_{C}}{{\mathbf {t} }\bullet \delta {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet \delta {\mathbf {u} }dV}=\delta \left({\int \limits _{S_{C}}{{\mathbf {t} }\bullet {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet {\mathbf {u} }dV}}\right)\\=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet \delta {\mathbf {e} }}\right)dV}=\int \limits _{V_{C}}{\delta WdV}=\delta \int \limits _{V_{C}}{WdV}\\\end{array}}$

(1.2.8)

or

$\displaystyle \delta \left({\int \limits _{S_{C}}{{\mathbf {t} }\bullet {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet {\mathbf {u} }dV}-W}\right)=\delta A=-\delta \Pi =0$

(1.2.9)

where $\displaystyle A$ is the total work done by external forces (surface and body forces) and internal forces (elastic forces) and $\displaystyle \Pi$is the total potential of the system.

$\displaystyle \int \limits _{S_{C}}{{\mathbf {t} }\bullet {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet {\mathbf {u} }dV}-W=A=-\Pi$

(1.2.10)

This equality means that the potential energy has an extremum value.

The first law of thermodynamics

First calculate the power input:

$\displaystyle {P_{input}}=\int \limits _{S_{C}}{{\mathbf {t} }\bullet {\mathbf {v} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {\mathbf {b} }\bullet {\mathbf {v} }dV}=\int \limits _{V_{C}}{\underbrace {\left({{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}\,\,{\mathbf {+} }\,\rho {\mathbf {b} }}\right)} _{\rho {\frac {D{\mathbf {v} }}{Dt}}}\bullet {\mathbf {v} }dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\boldsymbol {\nabla }}{\mathbf {v} }}\right)dV}$

$\displaystyle =\int \limits _{V_{C}}{\rho {\frac {D{\mathbf {v} }}{Dt}}\bullet {\mathbf {v} }dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}=\int \limits _{V_{C}}{{\frac {1}{2}}\rho {\frac {D}{Dt}}\left({{\mathbf {v} }\bullet {\mathbf {v} }}\right)dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}$

$\displaystyle {\begin{array}{l}={\frac {D}{Dt}}\int \limits _{{V_{m}}\left(t\right)}{{\frac {1}{2}}\rho {\mathbf {v} }\bullet {\mathbf {v} }dV}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}={\frac {DK}{Dt}}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}\end{array}}$

(1.3.1)

Here $\displaystyle K$is the kinetic energy of the system and$\displaystyle \mathbf {D}$ is the rate of deformation tensor:

$\displaystyle {\mathbf {D} }={\frac {1}{2}}\left[{{{\left({{\boldsymbol {\nabla }}{\mathbf {v} }}\right)}^{T}}+{\boldsymbol {\nabla }}{\mathbf {v} }}\right]$

(1.3.2)

Second, calculate the heat input:

$\displaystyle {Q_{input}}=-\int \limits _{S_{C}}{{\mathbf {q} }\bullet {\mathbf {n} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {q_{s}}dV}$

(1.3.3)

where $\displaystyle {\mathbf {q} }$ is the heat flux vector per unit area and $\displaystyle {q}_{s}$ is the internal heat supply per unit mass. The first law of thermodynamics said that when a closed system is carried through a cycle and returned to its initial state:

$\displaystyle \int \limits _{cycle}{\left[{{P_{input}}+{Q_{input}}}\right]dt}=\int \limits _{cycle}{dE}=0\to dE=\left[{{P_{input}}+{Q_{input}}}\right]dt\leftrightarrow {\frac {dE}{dt}}=\left[{{P_{input}}+{Q_{input}}}\right]$

(1.3.4)

Where $\displaystyle E$ is the total enery of the system. Generally, decompose the total energy into kinetic energy plus internal energy, so

$\displaystyle E=\int \limits _{V_{C}}{\left({{\frac {1}{2}}\rho {\mathbf {v} }\bullet {\mathbf {v} }+\rho u}\right)dV}=K+\int \limits _{V_{C}}{\rho udV}$

(1.3.5)

with $\displaystyle u$ is internal energy per unit mass (stored elastic energy) We are now in a position to write the first law of thermodynamics:

$\displaystyle {\begin{array}{l}{\frac {DE}{Dt}}={\frac {D}{Dt}}\int \limits _{{V_{m}}\left(t\right)}{\left({{\frac {1}{2}}\rho {\mathbf {v} }\bullet {\mathbf {v} }+\rho u}\right)dV}={\frac {DK}{Dt}}+\int \limits _{V_{C}}{\rho {\frac {Du}{Dt}}dV}\end{array}}$

$\displaystyle {\begin{array}{l}={P_{input}}+{Q_{input}}={\frac {DK}{Dt}}+\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}-\int \limits _{S_{C}}{{\mathbf {q} }\bullet {\mathbf {n} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {q_{s}}dV}\end{array}}$

(1.3.6)

$\displaystyle \to \int \limits _{V_{C}}{\rho {\frac {Du}{Dt}}dV}=\int \limits _{V_{C}}{tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)dV}-\int \limits _{S_{C}}{{\mathbf {q} }\bullet {\mathbf {n} }dS}+\,\,\,\int \limits _{V_{C}}{\rho {q_{s}}dV}$

$\displaystyle {\begin{array}{l}\leftrightarrow \rho {\frac {Du}{Dt}}=tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)-{\boldsymbol {\nabla }}\bullet {\mathbf {q} }+\,\,\,\rho {q_{s}}\end{array}}$

(1.3.7)

The second law of thermodynamics

For reversible processes,

$\displaystyle \int \limits _{cycle}{{\frac {Q_{input}}{\theta }}dt}=0\to dS={\frac {Q_{input}}{\theta }}dt\leftrightarrow {\frac {dS}{dt}}={\frac {Q_{input}}{\theta }}$

(1.4.1)

Where $\displaystyle S$ is the entropy of the system and $\displaystyle \theta$ is the absolute temperature. Expressing entropy of the system through its specific entropy $\displaystyle \eta$ (entropy per unit mass)

$\displaystyle S=\int \limits _{V_{C}}{\rho \eta dV}\to {\frac {D}{Dt}}\int \limits _{{V_{m}}\left(t\right)}{\rho \eta dV}=-\int \limits _{S_{C}}{{\frac {{\mathbf {q} }\bullet {\mathbf {n} }}{\theta }}dS}+\,\,\,\int \limits _{V_{C}}{{\frac {\rho {q_{s}}}{\theta }}dV}$

(1.4.2)

for reversible processes In irreversible processes, there is also internal entropy production from dissipative processes like internal friction, hence for general processes:

$\displaystyle {\frac {D}{Dt}}\int \limits _{{V_{m}}\left(t\right)}{\rho \eta dV}\geq -\int \limits _{S_{C}}{{\frac {{\mathbf {q} }\bullet {\mathbf {n} }}{\theta }}dS}+\,\,\,\int \limits _{V_{C}}{{\frac {\rho {q_{s}}}{\theta }}dV}\leftrightarrow \rho {\frac {D\eta }{Dt}}\geq {\frac {\rho {q_{s}}}{\theta }}-{\boldsymbol {\nabla }}\bullet \left({\frac {\mathbf {q} }{\theta }}\right)\,\,\,$

(1.4.3)

Thermoelasticity postulates

The second law of thermodynamics is

$\displaystyle \rho {\frac {D\eta }{Dt}}\geq {\frac {\rho {q_{s}}}{\theta }}-{\boldsymbol {\nabla }}\bullet \left({\frac {\mathbf {q} }{\theta }}\right)={\frac {\rho {q_{s}}}{\theta }}-{\frac {{\boldsymbol {\nabla }}\bullet {\mathbf {q} }}{\theta }}+{\frac {{\mathbf {q} }\bullet {\boldsymbol {\nabla }}\theta }{\theta ^{2}}}$

(1.5.1)

The last tebold in the above equation corresponds to the part of entropy production rate due to irreverible heat conduction in the presence of a temperature gradient (irreversible since heat only spontaneously flows from higher to lower temperature or $\displaystyle {\mathbf {q} }\bullet {\boldsymbol {\nabla }}\theta \leq 0$) Now assume that purely elastic behavior is fully reversible, then the inequality becomes an equality and$\displaystyle {\mathbf {q} }\bullet {\boldsymbol {\nabla }}\theta =0$ :

$\displaystyle \rho {\frac {D\eta }{Dt}}={\frac {\rho {q_{s}}}{\theta }}-{\frac {{\boldsymbol {\nabla }}\bullet {\mathbf {q} }}{\theta }}\to \rho {\frac {Du}{Dt}}=tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)-{\boldsymbol {\nabla }}\bullet {\mathbf {q} }+\,\,\,\rho {q_{s}}=tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)+\rho \theta {\frac {D\eta }{Dt}}$

(1.5.2)

Hence, internal energy is a function of entropy and deformations

$\displaystyle u=u\left({{\mathbf {E} },\eta }\right)$

(1.5.3)

Consider two cases: i) When deformation is isentropic, $\displaystyle {\frac {D\eta }{Dt}}=0$ , which is equivalent to adiabatic (no heat flow, valid in rapid deformations)

$\displaystyle \rho {\frac {Du}{Dt}}=tr\left({{\boldsymbol {\sigma }}\bullet {\mathbf {D} }}\right)$

(1.5.4)

but

$\displaystyle {\mathbf {D=} }{{\mathbf {F} }^{-T}}\bullet {\frac {D{\mathbf {E} }}{Dt}}\bullet {{\mathbf {F} }^{-1}}$

(1.5.5)

$\displaystyle \to \rho {\frac {Du}{Dt}}=tr\left({{\boldsymbol {\sigma }}\bullet {{\mathbf {F} }^{-T}}\bullet {\frac {D{\mathbf {E} }}{Dt}}\bullet {{\mathbf {F} }^{-1}}}\right)=tr\left({{{\mathbf {F} }^{-1}}\bullet {\boldsymbol {\sigma }}\bullet {{\mathbf {F} }^{-T}}\bullet {\frac {D{\mathbf {E} }}{Dt}}}\right)=tr\left({{\frac {1}{J}}{\boldsymbol {\tilde {\sigma }}}\bullet {\frac {D{\mathbf {E} }}{Dt}}}\right)={\frac {1}{J}}tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet {\frac {D{\mathbf {E} }}{Dt}}}\right)$

(1.5.6)

$\displaystyle \leftrightarrow J\rho {\frac {Du}{Dt}}={\rho _{0}}{\frac {Du}{Dt}}=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet {\frac {D{\mathbf {E} }}{Dt}}}\right)\leftrightarrow d\left({{\rho _{0}}u}\right)=dW=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet d{\mathbf {E} }}\right)$

(1.5.7)

Because

$\displaystyle \eta =const\to u=u\left({{\mathbf {E} },\eta }\right)=u\left({\mathbf {E} }\right)$

(1.5.8)

$\displaystyle \to W=W\left({\mathbf {E} }\right)\to dW=tr\left({{\frac {\partial W}{\partial {\mathbf {E} }}}\bullet d{\mathbf {E} }}\right)=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet d{\mathbf {E} }}\right)\to {\boldsymbol {\tilde {\sigma }}}={\frac {\partial W}{\partial {\mathbf {E} }}}$

(1.5.9)

ii) When deformation is isotheboldal, then $\displaystyle {\frac {d\theta }{dt}}=0$ (no temperature change, slow deformations)

For reversible processes we still have:

$\displaystyle \leftrightarrow {\rho _{0}}{\frac {D\left({u-\theta \eta }\right)}{Dt}}=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet {\frac {D{\mathbf {E} }}{Dt}}}\right)\leftrightarrow d\left[{{\rho _{0}}\left({u-\theta \eta }\right)}\right]=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet d{\mathbf {E} }}\right)$

(1.5.10)

Define the Helmholz free energy per unit mass:

$\displaystyle \psi \left({{\mathbf {E} },\theta }\right)=u-\theta \eta \to d\left({{\rho _{0}}\psi }\right)=dW=tr\left({{\boldsymbol {\tilde {\sigma }}}\bullet d{\mathbf {E} }}\right)$

(1.5.11)

Similarly, due to

$\displaystyle \theta =const\to \psi =\psi \left({{\mathbf {E} },\theta }\right)=\psi \left({\mathbf {E} }\right)$

(1.5.12)

and we end up with the same result

$\displaystyle {\boldsymbol {\tilde {\sigma }}}={\frac {\partial W}{\partial {\mathbf {E} }}}$

(1.5.13)

Thus in the cases of very rapid or very slow deformations, the elastic constitutive equations take the fobold …., with the strain energy density function $\displaystyle W$ being a function depending on $\displaystyle \mathbf {E}$only (for other types of deformation, $\displaystyle W$ would also have to depend on $\displaystyle \eta$ or $\displaystyle \theta$.

Refer to Pearson's book (give detailed ref and page number) who said that for processes with intermediate deformation rates, i.e., between very slow and very fast, one can still use (1.5.13) as a good approximation.

Eml5526.s11 16:26, 3 June 2011 (UTC)

General theory of elastic stability

Consider an arbitrary elastic body which is free from stress (state I.) By application of load or of heat, the body changes its shape and position (state II.) The material particlee has moved from the point, initially $\displaystyle {\mathbf {X} }$ to the point $\displaystyle {\mathbf {x} }$. The displacement vector is:

$\displaystyle {\mathbf {v} }={\mathbf {x} }-{\mathbf {X} }$

(2.1)

The Lagrangian strain tensor is:

$\displaystyle {\mathbf {E} }={\frac {1}{2}}\left[{\nabla {\mathbf {v} }+{{\left({\nabla {\mathbf {v} }}\right)}^{T}}+\nabla {\mathbf {v} }\bullet {{\left({\nabla {\mathbf {v} }}\right)}^{T}}}\right]$

(2.2)

Let $\displaystyle U$ be the internal energy per unit mass and assume constant entropy. Then, $\displaystyle U=U(\mathbf {E} )$ and

$\displaystyle {\mathbf {\tilde {\boldsymbol {\sigma }}} }={\frac {\partial \left({{\rho _{0}}U}\right)}{\partial {\mathbf {E} }}}={\rho _{0}}{\frac {\partial U}{\partial {\mathbf {E} }}}\to {\mathbf {\boldsymbol {\sigma }} }={\frac {1}{J}}{\mathbf {F} }\bullet {\mathbf {\tilde {\boldsymbol {\sigma }}} }\bullet {{\mathbf {F} }^{T}}={\frac {\rho }{\rho _{0}}}{\mathbf {F} }\bullet {\rho _{0}}{\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}=\rho {\mathbf {F} }\bullet {\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}$

(2.3)

Remember to define ${\tilde {\boldsymbol {\sigma }}}$ as the second Piola-Kirchhoff stress, which is usually denoted as ${\boldsymbol {S}}$.

Eml5526.s11 15:55, 3 June 2011 (UTC)

It is now required to analyze the stability of the body in its defobolded state II. The stability condition is “for each infinitesimal displacement which is compatible with the boundary conditions, the work done by the surface and body forces does not exceed that absorbed as an increase in internal energy.” The body forces per unit mass assumed to be constant. Consider two cases:

The surface foces are dead loads

The surface fores vary neither in magnitude nor in direction during trial displacement. It means that the total traction vector is the same irrespective of the direction of the area. The work done by the body and surface forces in a trial displacement $\displaystyle \mathbf {u}$ from state II is:

$\displaystyle {A_{e}}=\int \limits _{S_{II}}{{\mathbf {t} }\bullet {\mathbf {u} }dS}+\,\,\,\int \limits _{V_{II}}{\rho {\mathbf {b} }\bullet {\mathbf {u} }dV}=\int \limits _{V_{II}}{tr\left({{\mathbf {\boldsymbol {\sigma }} }\bullet \nabla {\mathbf {u} }}\right)dV\,}=\int \limits _{V_{II}}{tr\left({\rho {\mathbf {F} }\bullet {\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}\bullet \nabla {\mathbf {u} }}\right)dV\,}$

$\displaystyle =\int \limits _{V_{II}}{\rho tr\left({{\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}\bullet \nabla {\mathbf {u} }\bullet {\mathbf {F} }}\right)dV\,}=\int \limits _{V_{II}}{\rho tr\left({{\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\frac {\partial {\mathbf {x} }}{\partial {\mathbf {X} }}}}\right)dV\,}$

$\displaystyle =\int \limits _{V_{II}}{\rho tr\left({{\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)dV\,}$

(2.1.1)

The increase in internal energy is:

$\displaystyle \Omega =\int \limits _{V_{II}}{\rho \left({{U^{'}}-U}\right)dV}$

(2.1.2)

To be consistent with the notation used above, $U$ should be written in lowercase $u$ (which is energy per unit volume per unit mass, i.e., the specific energy). In standard books like Malvern 1969, the notation "$e$" is used. Eml5526.s11 15:38, 10 June 2011 (UTC)

The stability condition is:

$\displaystyle {A_{e}}-\Omega =\int \limits _{V_{II}}{\rho \left[{tr\left({{\frac {\partial U}{\partial {\mathbf {E} }}}\bullet {{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)-\left({{U^{'}}-U}\right)}\right]dV\,}\leq 0$

(2.1.3)

That is, stability is maintained if the external work does NOT surpass (less than or equal to) the increase in internal work. In other words, instability occurs when the external work is greater than the increase in internal work.

Eml5526.s11 15:48, 3 June 2011 (UTC)

Here the internal energy is:

$\displaystyle {U^{'}}=U\left({{\mathbf {E} }^{'}}\right),\,\,U=U\left({\mathbf {E} }\right)$

(2.1.4)

where

$\displaystyle {{\mathbf {E} }^{'}}={\frac {1}{2}}\left\{{{\frac {\partial }{\partial {\mathbf {X} }}}\left({{\mathbf {v} }+{\mathbf {u} }}\right)+{{\left[{{\frac {\partial }{\partial {\mathbf {X} }}}\left({{\mathbf {v} }+{\mathbf {u} }}\right)}\right]}^{T}}+{{\left[{{\frac {\partial }{\partial {\mathbf {X} }}}\left({{\mathbf {v} }+{\mathbf {u} }}\right)}\right]}^{T}}\bullet {\frac {\partial }{\partial {\mathbf {X} }}}\left({{\mathbf {v} }+{\mathbf {u} }}\right)}\right\}$ $\displaystyle ={\mathbf {E} }+{\frac {1}{2}}\left[{{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{{\left({\frac {\partial {\mathbf {v} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {v} }}{\partial {\mathbf {X} }}}}\right]$

$\displaystyle ={\mathbf {E} }+{\frac {1}{2}}\left[{\left({\underbrace {{\mathbf {I} }+{{\left({\frac {\partial {\mathbf {v} }}{\partial {\mathbf {X} }}}\right)}^{T}}} _{{\mathbf {F} }^{T}}}\right)\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet \underbrace {\left({{\mathbf {I} }+{\frac {\partial {\mathbf {v} }}{\partial {\mathbf {X} }}}}\right)} _{\mathbf {F} }+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right]$

$\displaystyle ={\mathbf {E} }+{\frac {1}{2}}\left[{{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\mathbf {F} }+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right]={\mathbf {E} }+sym\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)+{\frac {1}{2}}{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}$

(2.1.5)

Using Taylor’s expansion of internal energy around state II to get:

$\displaystyle {U^{'}}-U=U\left({{\mathbf {E} }^{'}}\right)-U\left({\mathbf {E} }\right)={\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|_{II}}\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)+{\frac {1}{2}}{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{II}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}+\cdots$

$\displaystyle =tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet \left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)}\right)+{\frac {1}{2}}{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{II}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}+\cdots$

(2.1.6)

$\displaystyle =tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet sym\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)}\right)+{\frac {1}{2}}tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)+{\frac {1}{2}}{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{II}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}+\cdots$

$\displaystyle =tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet \left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)}\right)+{\frac {1}{2}}tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)+{\frac {1}{2}}{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{II}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}+\cdots$

(2.1.7)

The stability condition reduces:

$\displaystyle \int \limits _{V_{II}}{\rho \left[{tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)+{{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|}_{II}}{{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)}^{2}}+\cdots }\right]dV\geq \,}0$

(2.1.8)

If we write

$\displaystyle {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}={\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\frac {\partial {\mathbf {x} }}{\partial {\mathbf {X} }}}={\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {F} }$

(2.1.9)

and convert

$\displaystyle \rho tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)=\rho tr\left({{{\left.{{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\bullet {\frac {\partial U}{\partial {\mathbf {E} }}}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}}\right)$

$\displaystyle =\rho tr\left({{{\left.{{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {F} }\bullet {\frac {\partial U}{\partial {\mathbf {E} }}}}\right|}_{II}}\bullet {{\mathbf {F} }^{T}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)=tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)$

(2.1.10)

then alternative fobold of stability condition is:

The intermediate step is: ${A_{e}}-\Omega =-{\frac {1}{2}}\int \limits _{V_{II}}{\rho \left[tr\left({{{\left.{\frac {\partial U}{\partial {\mathbf {E} }}}\right|}_{II}}\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\right)}^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)+{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{II}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}+\cdots \right]dV}\leq 0$

Eml5526.s11 16:12, 3 June 2011 (UTC)

$\displaystyle \int \limits _{V_{II}}{\left[{tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)+{{\left.{\rho {\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}}\right|}_{II}}{{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)}^{2}}+\cdots }\right]dV\geq \,}0$

(2.1.11)

Calculating the second term explicitly:

$\displaystyle {\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}{\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)^{2}}=\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right){\mathbf {:} }{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}{\mathbf {:} }\left({{{\mathbf {E} }^{'}}-{\mathbf {E} }}\right)$

$\displaystyle \approx sym\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right){\mathbf {:} }{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}{\mathbf {:} }sym\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)=\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right){\mathbf {:} }{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}{\mathbf {:} }\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)$

(2.1.12)

And the final expression for stability condition is:

$\displaystyle \int \limits _{V_{II}}{\left[{tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)+\rho \left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right){\mathbf {:} }{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}{\mathbf {:} }\left({{{\mathbf {F} }^{T}}\bullet {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}}\right)}\right]dV\geq \,}0$

(2.1.13)

In indicial notation, it is

$\displaystyle \int \limits _{V_{II}}{\left[{{\sigma _{ij}}{\frac {\partial {u_{k}}}{\partial {x_{i}}}}{\frac {\partial {u_{k}}}{\partial {x_{j}}}}+\rho {\frac {{\partial ^{2}}U}{\partial {E_{ij}}\partial {E_{pq}}}}{\frac {\partial {x_{k}}}{\partial {X_{i}}}}{\frac {\partial {u_{k}}}{\partial {X_{j}}}}{\frac {\partial {x_{l}}}{\partial {X_{p}}}}{\frac {\partial {u_{l}}}{\partial {X_{q}}}}}\right]dV\geq \,}0$

(2.1.14)

Here all quantities are calculated for state II. Using Taylor’s expansion again to write the second tebold:

$\displaystyle {\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}={\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{I}}+{\left.{\frac {{\partial ^{3}}U}{\partial {{\mathbf {E} }^{3}}}}\right|_{I}}\left({{\mathbf {E} }-{{\mathbf {E} }_{I}}}\right)+\cdots \approx {\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{I}}$

$\displaystyle \rho {\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\approx {\rho _{0}}{\left.{\frac {{\partial ^{2}}U}{\partial {{\mathbf {E} }^{2}}}}\right|_{I}}={\left.{\frac {{\partial ^{2}}\left({{\rho _{0}}U}\right)}{\partial {{\mathbf {E} }^{2}}}}\right|_{I}}={{\mathbf {C} }_{I}}$

(2.1.15)

Moreover, assuming the defoboldation between states I and II are small (the displacements $\displaystyle \mathbf {u}$ are not necessarily small), so there are no differences between positions $\displaystyle \mathbf {x}$ and $\displaystyle \mathbf {X}$:

$\displaystyle {\mathbf {F=} }{\frac {\partial {\mathbf {x} }}{\partial {\mathbf {X} }}}\approx {\mathbf {I} },\,\,{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {X} }}}\approx {\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}},\,\,{{\mathbf {C} }_{I}}\approx {{\mathbf {C} }_{II}}\equiv {\mathbf {c} }$

(2.1.16)

Thus, stability condition becomes:

$\displaystyle \int \limits _{V_{II}}{\left[{tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)+{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}{\mathbf {:c:} }{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}}\right]dV\geq \,}0$

$\displaystyle \leftrightarrow \int \limits _{V_{II}}{\left[{tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)+sym\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right){\mathbf {:c:} }sym\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}\right]dV\geq \,}0$

$\displaystyle \leftrightarrow \int \limits _{V_{II}}{\left[{tr\left({{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\bullet {\mathbf {\boldsymbol {\sigma }} }\bullet {{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right)+{\mathbf {e:c:e} }}\right]dV\geq \,}0$

(2.1.17)

where $\displaystyle \mathbf {e}$ is defined (in state II):

$\displaystyle {\mathbf {e} }={\frac {1}{2}}\left[{\nabla {\mathbf {u} }+{{\left({\nabla {\mathbf {u} }}\right)}^{T}}}\right]={\frac {1}{2}}\left[{{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}+{{\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)}^{T}}}\right]$

(2.1.18)

or in the indical notation,

$\displaystyle \int \limits _{V_{II}}{\left[{{\sigma _{ij}}{\frac {\partial {u_{k}}}{\partial {x_{i}}}}{\frac {\partial {u_{k}}}{\partial {x_{j}}}}+{c_{mnpq}}{e_{mn}}{e_{pq}}}\right]dV\geq \,}0$

(2.1.19)

For isotropic media,

$\displaystyle {c_{mnpq}}=\lambda {\delta _{mn}}{\delta _{pq}}+\mu \left({{\delta _{mp}}{\delta _{nq}}+{\delta _{mq}}{\delta _{np}}}\right)$

(2.1.20)

$\displaystyle \lambda$ and $\displaystyle \mu$ are Lame’s constants. This relation reduces the stability criterion to:

$\displaystyle \int \limits _{V_{II}}{\left[{{\sigma _{ij}}{\frac {\partial {u_{k}}}{\partial {x_{i}}}}{\frac {\partial {u_{k}}}{\partial {x_{j}}}}+\lambda {e_{mm}}{e_{nn}}+2\mu {e_{mn}}{e_{mn}}}\right]dV\geq \,}0$

(2.1.21)

Can you deduce from (2.1.21) that $\displaystyle \lambda +\mu >0$ ? Since this condition is used later on to show that the Poisson's ratio of a stable composite material can be negative. Egm6321.f11 15:12, 24 June 2011 (UTC)

The surface foces are pressure loading

We assume that the system is still conservative so the total work done by pressure forces is independent of the path. Hence, let the trial displacements $\displaystyle \mathbf {u}$ grow at a constant rate, i.e at the time $\displaystyle t$, displacements are $\displaystyle \mathbf {u} t$. The work done in the interval from $\displaystyle t=0$ to $\displaystyle t=1$ is:

$\displaystyle A_{e}^{1}=\int \limits _{0}^{1}{dt}\int \limits _{S_{III}}{\left({-P{{\mathbf {n} }_{t}}}\right)}\bullet {\frac {d}{dt}}\left({{\mathbf {u} }t}\right)d{s_{t}}=-P\int \limits _{0}^{1}{dt}\int \limits _{S_{III}}{{\mathbf {u} }\bullet {{\mathbf {n} }_{t}}d{s_{t}}}$

(2.2.1)

You need to define the surface $S_{III}$, which is the perturbed configuration from the surface $S_{II}$ at equilibrium. Eml5526.s11 14:26, 10 June 2011 (UTC)

There are 3 configurations: Reference configuration ${\mathcal {B}}$, the current equilibrium configuration ${\mathcal {B}}_{0}$, the perturbed configuration ${\mathcal {B}}_{t}$. The modern notation for different quantities, e.g., stresses, strains, etc. is as follows. In the reference configuration ${\mathcal {B}}$, uppercase letters are used, e.g., ${\boldsymbol {P}}$ (first Piola-Kirchhoff stress tensor), ${\boldsymbol {E}}$ (Green-Lagrange strain tensor), ${\boldsymbol {N}}$ (normal to boundary surface $\partial {\mathcal {B}}$), etc. In the current equilibrium configuration ${\mathcal {B}}_{0}$, lowercase letters are used, e.g., ${\boldsymbol {\sigma }}_{0}$ (Cauchy stress tensor), ${\boldsymbol {\epsilon }}_{0}$ (small strain tensor), ${\boldsymbol {n}}_{0}$ (normal to boundary surface $\partial {\mathcal {B}}_{0}$), etc. In the perturbed configuration ${\mathcal {B}}_{t}$, lowercase letters are used, e.g., ${\boldsymbol {\sigma }}_{t}$ (Cauchy stress tensor), ${\boldsymbol {\epsilon }}_{t}$ (small strain tensor), ${\boldsymbol {n}}_{t}$ (normal to boundary surface $\partial {\mathcal {B}}_{0}$), etc. Eml5526.s11 14:26, 10 June 2011 (UTC)

where $\displaystyle \mathbf {n} _{t}$ is the normal vector of the area element $\displaystyle ds_{t}$ in the spatial configuration and is related to the normal vector $\displaystyle n$ of the area element $\displaystyle ds$ in the reference configuration as follows:

$\displaystyle {{\mathbf {n} }_{t}}d{s_{t}}=\det \left({{\mathbf {F} }_{t}}\right){\mathbf {F} }_{t}^{-T}\bullet {\mathbf {n} }ds$

(2.2.2)

where $\displaystyle {\mathbf {F} _{t}}$ is the deformation gradient

$\displaystyle {{\mathbf {F} }_{t}}={\frac {\partial {{\mathbf {x} }_{t}}}{\partial {\mathbf {x} }}}={\frac {\partial \left({{\mathbf {x} }+{\mathbf {u} }t}\right)}{\partial {\mathbf {x} }}}={\mathbf {I} }+t{\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}={\mathbf {I} }+t{\mathbf {u} }{\overleftarrow {\nabla }}$

(2.2.3)

and

$\displaystyle {\mathbf {u} }{\overleftarrow {\nabla }}={\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}$

(2.2.4)

From this, we can determine:

$\displaystyle \det \left({{\mathbf {F} }_{t}}\right)=1+t{I_{1}}+{t^{2}}{I_{2}}+{t^{3}}{I_{3}}$

(2.2.5)

where

$\displaystyle {I_{1}}=tr\left({{\mathbf {u} }{\overleftarrow {\nabla }}}\right),\,\,{I_{2}}={\frac {1}{2}}\left[{t{r^{2}}\left({{\mathbf {u} }{\overleftarrow {\nabla }}}\right)-tr\left({{\mathbf {u} }{\overleftarrow {\nabla }}\bullet {\mathbf {u} }{\overleftarrow {\nabla }}}\right)}\right],\,\,{I_{3}}=\det \left({{\mathbf {u} }{\overleftarrow {\nabla }}}\right)$

(2.2.6)

We assume infinitesimal deformation,

$\displaystyle \left|{{\mathbf {u} }{\overleftarrow {\nabla }}}\right|<<1$

(2.2.7)

so

$\displaystyle {\mathbf {F} }_{t}^{-1}={\left({{\mathbf {I} }+t{\mathbf {u} }{\overleftarrow {\nabla }}}\right)^{-1}}\approx {\mathbf {I} }-t{\mathbf {u} }{\overleftarrow {\nabla }}+{t^{2}}{\left({{\mathbf {u} }{\overleftarrow {\nabla }}}\right)^{2}}-{t^{3}}{\left({{\mathbf {u} }{\overleftarrow {\nabla }}}\right)^{3}}+\cdots$

(2.2.8)

or

$\displaystyle {\mathbf {F} }_{t}^{-T}={\left({{\mathbf {I} }+t{\mathbf {u} }{\overleftarrow {\nabla }}}\right)^{-T}}\approx {\mathbf {I} }-t\nabla {\mathbf {u} }+{t^{2}}{\left({\nabla {\mathbf {u} }}\right)^{2}}-{t^{3}}{\left({\nabla {\mathbf {u} }}\right)^{3}}+\cdots$

(2.2.9)

here

$\displaystyle \nabla {\mathbf {u} }={\left({\frac {\partial {\mathbf {u} }}{\partial {\mathbf {x} }}}\right)^{T}}$

(2.2.10)

The integrand becomes:

$\displaystyle {\mathbf {u} }\bullet {{\mathbf {n} }_{t}}d{s_{t}}=\det \left({{\mathbf {F} }_{t}}\right){\mathbf {u} }\bullet {\mathbf {F} }_{t}^{-T}\bullet {\mathbf {n} }ds\approx \left({1+t{I_{1}}+{t^{2}}{I_{2}}+{t^{3}}{I_{3}}}\right){\mathbf {u} }\bullet \left({{\mathbf {I} }-t\nabla {\mathbf {u} }+{t^{2}}{{\left({\nabla {\mathbf {u} }}\right)}^{2}}+\cdots }\right)\bullet {\mathbf {n} }ds$

$\displaystyle ={\mathbf {u} }\bullet \left\{{{\mathbf {I} }+t\left({{I_{1}}{\mathbf {I} }-\nabla {\mathbf {u} }}\right)+{t^{2}}\left[{{{\left({\nabla {\mathbf {u} }}\right)}^{2}}-{I_{1}}\nabla {\mathbf {u} }+{I_{2}}{\mathbf {I} }}\right]+\cdots }\right\}\bullet {\mathbf {n} }ds$

(2.2.11)

And the work done by pressure forces is:

$\displaystyle A_{e}^{1}=-P\int \limits _{0}^{1}{dt}\int \limits _{S_{II}}{\det \left({{\mathbf {F} }_{t}}\right){\mathbf {u} }\bullet {\mathbf {F} }_{t}^{-T}\bullet {\mathbf {n} }ds}=-P\int \limits _{S_{II}}{{\mathbf {u} }\bullet \int \limits _{0}^{1}{\left\{{{\mathbf {I} }+t\left({{I_{1}}{\mathbf {I} }-\nabla {\mathbf {u} }}\right)+{t^{2}}\left[{{{\left({\nabla {\mathbf {u} }}\right)}^{2}}-{I_{1}}\nabla {\mathbf {u} }+{I_{2}}{\mathbf {I} }}\right]+\cdots }\right\}dt}\bullet {\mathbf {n} }ds}$

$\displaystyle =-P\int \limits _{S_{II}}{{\mathbf {u} }\bullet \left\{{{\mathbf {I} }+{\frac {1}{2}}\left({{I_{1}}{\mathbf {I} }-\nabla {\mathbf {u} }}\right)+{\frac {1}{3}}\left[{{{\left({\nabla {\mathbf {u} }}\right)}^{2}}-{I_{1}}\nabla {\mathbf {u} }+{I_{2}}{\mathbf {I} }}\right]+\cdots }\right\}\bullet {\mathbf {n} }ds}\approx -P\int \limits _{S_{II}}{{\mathbf {u} }\bullet \left[{{\mathbf {I} }+{\frac {1}{2}}\left({{I_{1}}{\mathbf {I} }-\nabla {\mathbf {u} }}\right)}\right]\bullet {\mathbf {n} }ds}$

(2.2.12)

Note that, Pearson got the above expression exactly as follows, , p. 141-142, using indical notation, , p. 16:

$\displaystyle {\left({{\mathbf {n} }_{i}}\right)_{t}}{\left({ds}\right)_{t}}={\frac {1}{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{\frac {\partial \left({{x_{j}}+t{u_{j}}}\right)}{\partial {x_{m}}}}{\frac {\partial \left({{x_{k}}+t{u_{k}}}\right)}{\partial {x_{n}}}}{n_{p}}ds={\frac {1}{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{\frac {\partial \left({{x_{j}}+t{u_{j}}}\right)}{\partial {x_{m}}}}{\frac {\partial \left({{x_{k}}+t{u_{k}}}\right)}{\partial {x_{n}}}}{n_{p}}ds$

$\displaystyle ={\frac {1}{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}\left({{\delta _{jm}}+t{\frac {\partial {u_{j}}}{\partial {x_{m}}}}}\right)\left({{\delta _{kn}}+t{\frac {\partial {u_{k}}}{\partial {x_{n}}}}}\right){n_{p}}ds={u_{i}}\left({{\delta _{ip}}+t{\frac {\partial {u_{k}}}{\partial {x_{k}}}}{\delta _{ip}}-t{\frac {\partial {u_{p}}}{\partial {x_{i}}}}+{\frac {1}{2}}{t^{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{\frac {\partial {u_{j}}}{\partial {x_{m}}}}{\frac {\partial {u_{k}}}{\partial {x_{n}}}}}\right){n_{p}}ds$

$\displaystyle =\left({{u_{i}}{n_{i}}+t{u_{i}}{n_{i}}{\frac {\partial {u_{k}}}{\partial {x_{k}}}}-t{u_{i}}{\frac {\partial {u_{p}}}{\partial {x_{i}}}}{n_{p}}+{\frac {1}{2}}{t^{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{u_{i}}{\frac {\partial {u_{j}}}{\partial {x_{m}}}}{\frac {\partial {u_{k}}}{\partial {x_{n}}}}{n_{p}}}\right)ds$

(2.2.13)

Plugging these formular into expression (2.2.1) to obtain:

$\displaystyle A_{e}^{1}=-P\int \limits _{0}^{1}{dt}\int \limits _{S_{III}}{{u_{i}}{{\left({{\mathbf {n} }_{i}}\right)}_{t}}{{\left({ds}\right)}_{t}}}=-P\int \limits _{S_{II}}ds\int \limits _{0}^{1}{\left({{u_{i}}{n_{i}}+t{u_{i}}{n_{i}}{\frac {\partial {u_{k}}}{\partial {x_{k}}}}-t{u_{i}}{\frac {\partial {u_{p}}}{\partial {x_{i}}}}{n_{p}}+{\frac {1}{2}}{t^{2}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{u_{i}}{\frac {\partial {u_{j}}}{\partial {x_{m}}}}{\frac {\partial {u_{k}}}{\partial {x_{n}}}}{n_{p}}}\right)}{dt}$

$\displaystyle =-P\int \limits _{S_{II}}{\left({{u_{i}}{n_{i}}+{\frac {1}{2}}{u_{i}}{n_{i}}{\frac {\partial {u_{k}}}{\partial {x_{k}}}}-{\frac {1}{2}}{u_{i}}{\frac {\partial {u_{p}}}{\partial {x_{i}}}}{n_{p}}+{\frac {1}{6}}{\varepsilon _{jki}}{\varepsilon _{mnp}}{u_{i}}{\frac {\partial {u_{j}}}{\partial {x_{m}}}}{\frac {\partial {u_{k}}}{\partial {x_{n}}}}{n_{p}}}\right)ds}$

(2.2.14)

Convert back to direct notation:

$\displaystyle A_{e}^{1}=-P\int \limits _{S_{II}}{\left\{{{\mathbf {u} }\bullet {\mathbf {n} }+{\frac {1}{2}}\left[{\left({\nabla \bullet {\mathbf {u} }}\right){\mathbf {u} }\bullet {\mathbf {n} }-{\mathbf {u} }\bullet \left({\nabla {\mathbf {u} }}\right)\bullet {\mathbf {n} }}\right]+{\frac {1}{6}}\left({{\mathbf {u} }\times {\mathbf {u} }{\overleftarrow {\nabla }}{\mathbf {:u} }{\overleftarrow {\nabla }}\times {\mathbf {n} }}\right)}\right\}ds}$

(2.2.15)

You may want to refer to Pearson's notation as the "component form" to distinguish from the "tensor form" in Eq.(2.2.15) (which you called the "direct" notation) Eml5526.s11 15:19, 10 June 2011 (UTC)

Eq.(2.2.15) is to be used in the stability condition expressed in Eq.(2.1.3). It is not clear yet how .... You want to explain this point clearly. Eml5526.s11 15:19, 10 June 2011 (UTC)

Elastic stability of composite materials including an inclusion and a coating

Consider the stability of an elastic body intially free of stresses. The formular (2.1.21) reduces:

Give references. Egm6321.f11 14:46, 17 June 2011 (UTC)

$\displaystyle \int \limits _{V}{\left[{2\mu {\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+\lambda {{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dV>}0$

(3.1)

Integrating the whole domain of the body will be calculated through the inclusion $\displaystyle V_{1}$ and the coating $\displaystyle V_{2}$ :

$\displaystyle \int \limits _{V_{1}}{\left[{2{\mu _{1}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{1}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dV}+\int \limits _{V_{2}}{\left[{2{\mu _{2}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{2}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dV>}0$

(3.2)

Introducing an arbitrary parameter $\displaystyle \beta$ , $\displaystyle 0\leq \beta \leq 1$ to vary the inclusion moduli which will turn out to be the full range for composite stability from positive definite to strongly elliptic.

$\displaystyle {\mu _{1}}=\beta {\mu _{1}}+\left({1-\beta }\right){\mu _{1}}$

(3.3)

In the composite literature, the above equation is called the "rule of mixture". In mathematics, the above equation is called the "convex combination" of $\displaystyle \mu _{1}$ and $\displaystyle \mu _{2}$. Why not doing also a convex combination of the lambda's, i.e., $\displaystyle \lambda _{1}$ and $\displaystyle \lambda _{2}$. OK, the above is not the rule of mixture of $\displaystyle \mu _{1}$ and $\displaystyle \mu _{2}$, but only rewritting $\displaystyle \mu _{1}$ in terms of the parameter $\displaystyle \beta$. Why ? Egm6321.f11 14:51, 17 June 2011 (UTC)

Assume the coating is thin, so the strains are constanst through it and the second volume integration becomes the product of the surface one and the thickness $\displaystyle t$ of the coating. The stability condition is:

$\displaystyle \int \limits _{V_{1}}{\left[{2{\mu _{1}}\left({1-\beta }\right){\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+2{\mu _{1}}\beta {\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{1}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dV}+\int \limits _{S_{2}}{t\left[{2{\mu _{2}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{2}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dS>}0$

(3.4)

Decompose the strain tensor into the deviatoric tensor $\displaystyle {{\mathbf {\boldsymbol {\varepsilon }} }^{'}}$ and use the Kelvin’s identity as follows

$\displaystyle {\mathbf {\boldsymbol {\varepsilon }} }={{\mathbf {\boldsymbol {\varepsilon }} }^{'}}+{\frac {1}{k}}tr\left({\mathbf {\boldsymbol {\varepsilon }} }\right){\mathbf {I} }$

(3.5)

$\displaystyle {\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }={{\mathbf {\boldsymbol {\varepsilon }} }^{'}}{\mathbf {:} }{{\mathbf {\boldsymbol {\varepsilon }} }^{'}}+{\frac {1}{k}}{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)^{2}}$

(3.6)

$\displaystyle {\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }={\mathbf {{\boldsymbol {\omega }}:{\boldsymbol {\omega }}} }+{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)^{2}}+{\boldsymbol {\nabla }}\bullet \left[{\left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\bullet {\mathbf {u} }-{\mathbf {u} }\left({{\boldsymbol {\nabla }}\bullet {\mathbf {u} }}\right)}\right]$

(3.7)

Introducing the first identity into the first term, the second into the second, all in the first integrand, to get (after resorting)

$\displaystyle \int \limits _{V_{1}}{\left\{{2{\mu _{1}}\left[{\left({1-\beta }\right){{\mathbf {\boldsymbol {\varepsilon }} }^{'}}{\mathbf {:} }{{\mathbf {\boldsymbol {\varepsilon }} }^{'}}+\beta {\mathbf {{\boldsymbol {\omega }}:{\boldsymbol {\omega }}} }}\right]+\left[{{\lambda _{1}}+{\frac {2{\mu _{1}}}{k}}\left({1+\left({k-1}\right)\beta }\right)}\right]{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right\}dV}+$

(3.8)

$\displaystyle +2{\mu _{1}}\beta \int \limits _{V_{1}}{{\boldsymbol {\nabla }}\bullet \left[{\left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\bullet {\mathbf {u} }-{\mathbf {u} }\left({{\boldsymbol {\nabla }}\bullet {\mathbf {u} }}\right)}\right]dV}+\int \limits _{S_{2}}{t\left[{2{\mu _{2}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{2}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]dS>}0$

(3.9)

Using the divergence theorem and the following identity

$\displaystyle {\mathbf {n} }\bullet \left[{\left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\bullet {\mathbf {u} }-{\mathbf {u} }\left({{\boldsymbol {\nabla }}\bullet {\mathbf {u} }}\right)}\right]=tr\left[{{\mathbf {u} }\times \left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\times {\mathbf {n} }}\right]$

(3.10)

to convert the second volume integration into the surface one and remember that the inclusion and the coating share the same surface $\displaystyle S$ to get the final expression for stability condition:

$\displaystyle \int \limits _{V_{1}}{\left\{{2{\mu _{1}}\left[{\left({1-\beta }\right){{\mathbf {\boldsymbol {\varepsilon }} }^{'}}{\mathbf {:} }{{\mathbf {\boldsymbol {\varepsilon }} }^{'}}+\beta {\mathbf {{\boldsymbol {\omega }}:{\boldsymbol {\omega }}} }}\right]+\left[{{\lambda _{1}}+{\frac {2{\mu _{1}}}{k}}\left({1+\left({k-1}\right)\beta }\right)}\right]{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right\}dV}+$

(3.11)

$\displaystyle +\int \limits _{S}{\left\{{t\left[{2{\mu _{2}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{2}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]+2{\mu _{1}}\beta tr\left[{{\mathbf {u} }\times \left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\times {\mathbf {n} }}\right]}\right\}dS>}0$

(3.12)

${\begin{matrix}\displaystyle \int \limits _{V_{1}}{\left\{{2{\mu _{1}}\left[{\left({1-\beta }\right){{\mathbf {\boldsymbol {\varepsilon }} }^{'}}{\mathbf {:} }{{\mathbf {\boldsymbol {\varepsilon }} }^{'}}+\beta {\mathbf {{\boldsymbol {\omega }}:{\boldsymbol {\omega }}} }}\right]+\left[{{\lambda _{1}}+{\frac {2{\mu _{1}}}{k}}\left({1+\left({k-1}\right)\beta }\right)}\right]{{\left({\operatorname {tr} {\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right\}dV}\\\displaystyle +\int \limits _{S}{\left\{{t\left[{2{\mu _{2}}{\mathbf {{\boldsymbol {\varepsilon }}:{\boldsymbol {\varepsilon }}} }+{\lambda _{2}}{{\left({tr{\mathbf {\boldsymbol {\varepsilon }} }}\right)}^{2}}}\right]+2{\mu _{1}}\beta {\rm {tr}}\left[{{\mathbf {u} }\times \left({{\mathbf {u} }{\boldsymbol {\nabla }}}\right)\times {\mathbf {n} }}\right]}\right\}dS>}0\end{matrix}}$

(3.11)

$\displaystyle {\overset {e=nel}{\underset {e=1}{\operatorname {\mathbb {A} } }}}$

(N)

Now we are in a position to analyze two cases.

Plane strain coated cylinder problem

In this case $\displaystyle k=2$ and express the surface integral in terms of polar coordinate $\displaystyle r,\theta$ and $\displaystyle dS=ad\theta$.

Remember in polar coordinate we have the followings:

$\displaystyle {\boldsymbol {\varepsilon }}={\varepsilon _{rr}}{{\mathbf {e} }_{r}}{{\mathbf {e} }_{r}}+{\varepsilon _{r\theta }}{{\mathbf {e} }_{r}}{{\mathbf {e} }_{\theta }}+{\varepsilon _{r\theta }}{{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{r}}+{\varepsilon _{\theta \theta }}{{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{\theta }}$

(3.13)

$\displaystyle {\overleftarrow {\nabla }}={\frac {\partial }{\partial r}}{{\mathbf {e} }_{r}}+{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{{\mathbf {e} }_{\theta }}$

(3.14)

$\displaystyle {\mathbf {u} }={u_{r}}{{\mathbf {e} }_{r}}+{u_{\theta }}{{\mathbf {e} }_{\theta }}$

(3.15)

$\displaystyle {\mathbf {u} }{\overleftarrow {\nabla }}={\frac {\partial {u_{r}}}{\partial r}}{{\mathbf {e} }_{r}}{{\mathbf {e} }_{r}}+\left({{\frac {1}{r}}{\frac {\partial {u_{r}}}{\partial \theta }}-{\frac {u_{\theta }}{r}}}\right){{\mathbf {e} }_{r}}{{\mathbf {e} }_{\theta }}+{\frac {\partial {u_{\theta }}}{\partial r}}{{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{r}}+\left({{\frac {1}{r}}{\frac {\partial {u_{\theta }}}{\partial \theta }}+{\frac {u_{r}}{r}}}\right){{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{\theta }}$

(3.16)

so

$\displaystyle {\boldsymbol {\varepsilon :\varepsilon }}=\varepsilon _{rr}^{2}+\varepsilon _{\theta \theta }^{2}+2\varepsilon _{r\theta }^{2}$

(3.17)

$\displaystyle tr\left({\boldsymbol {\varepsilon }}\right)={\varepsilon _{rr}}+{\varepsilon _{\theta \theta }}$

(3.18)

$\displaystyle \mathbf {u} \times \mathbf {u} {\overleftarrow {\boldsymbol {\nabla }}}\times \mathbf {n} =\left({{u}_{r}}{{\mathbf {e} }_{r}}+{{u}_{\theta }}{{\mathbf {e} }_{\theta }}\right)\times \left[{\frac {\partial {{u}_{r}}}{\partial r}}{{\mathbf {e} }_{r}}{{\mathbf {e} }_{r}}+{\frac {1}{r}}\left({\frac {\partial {{u}_{r}}}{\partial \theta }}-{{u}_{\theta }}\right){{\mathbf {e} }_{r}}{{\mathbf {e} }_{\theta }}+{\frac {\partial {{u}_{\theta }}}{\partial r}}{{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{r}}+{\frac {1}{r}}\left({\frac {\partial {{u}_{\theta }}}{\partial \theta }}+{{u}_{r}}\right){{\mathbf {e} }_{\theta }}{{\mathbf {e} }_{\theta }}\right]\times {{\mathbf {e} }_{r}}$

$\displaystyle =-{\frac {1}{r}}\left({u_{r}^{2}+u_{\theta }^{2}+{u_{r}}{\frac {\partial {u_{\theta }}}{\partial \theta }}-{u_{\theta }}{\frac {\partial {u_{r}}}{\partial \theta }}}\right){{\mathbf {e} }_{z}}{{\mathbf {e} }_{z}}$

(3.19)

$\displaystyle tr\left({{\mathbf {u} }\times {\mathbf {u} }{\overleftarrow {\nabla }}\times {\mathbf {n} }}\right)=-{\frac {1}{r}}\left({u_{r}^{2}+u_{\theta }^{2}+{u_{r}}{\frac {\partial {u_{\theta }}}{\partial \theta }}-{u_{\theta }}{\frac {\partial {u_{r}}}{\partial \theta }}}\right)$

(3.10)

Plugging those expressions above into the stability condition to get:

$\displaystyle \int \limits _{A_{1}}{\left\{{2{\mu _{1}}\left[{\left({1-\beta }\right){{\boldsymbol {\varepsilon }}^{'}}{\mathbf {:} }{{\boldsymbol {\varepsilon }}^{'}}+\beta {\mathbf {\omega :\omega } }}\right]+\left[{{\lambda _{1}}+\left({1+\beta }\right){\mu _{1}}}\right]{{\left({tr{\boldsymbol {\varepsilon }}}\right)}^{2}}}\right\}dA}+$

$\displaystyle +\int \limits _{0}^{2\pi }{\left\{{ta\left[{2{\mu _{2}}\left({\varepsilon _{rr}^{2}+\varepsilon _{\theta \theta }^{2}+2\varepsilon _{r\theta }^{2}}\right)+{\lambda _{2}}{{\left({{\varepsilon _{rr}}+{\varepsilon _{\theta \theta }}}\right)}^{2}}}\right]-2\beta {\mu _{1}}\left[{{{\left({u_{r}}\right)}^{2}}+{{\left({u_{\theta }}\right)}^{2}}+{u_{r}}{u_{\theta ,\theta }}-{u_{\theta }}{u_{r,\theta }}}\right]}\right\}d\theta }>0$

(3.21)

Employ a Fourier series representation for $\displaystyle u_{\theta }$

$\displaystyle {u_{\theta }}=\sum _{n=-\propto }^{\propto }{{c_{n}}{e^{in\theta }}}$

(3.22)

Below is a reformatting of the above equation. Note that the symbol $\displaystyle \propto$ means "proportional to", whereas the symbol $\displaystyle \infty$ is infinity. Egm6321.f11 15:02, 17 June 2011 (UTC)

$\displaystyle u_{\theta }=\sum _{n=-\infty }^{\infty }c_{n}e^{in\theta }$

(3.22)

where $\displaystyle c_{-n}={\bar {c}}_{n}$ (overbar is the complex conjugate) and no rigid body motion requires $\displaystyle c_{0}=c_{1}=0$. These and orthogonality of Fourier series show on the boundary $\displaystyle r=a$ that

$\displaystyle \int \limits _{0}^{2\pi }{{{\left({u_{\theta }}\right)}^{2}}d\theta =4\pi \sum \nolimits _{n=2}^{\propto }{{c_{n}}{{\bar {c}}_{n}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\int \limits _{0}^{2\pi }{{{\left({u_{\theta ,\theta }}\right)}^{2}}d\theta =4\pi \sum \nolimits _{n=2}^{\propto }{{n^{2}}{c_{n}}{{\bar {c}}_{n}}}}$

(3.23)

Hence,

$\displaystyle 4\int \limits _{0}^{2\pi }{{{\left({u_{\theta }}\right)}^{2}}d\theta }\leq \int \limits _{0}^{2\pi }{{{\left({u_{\theta ,\theta }}\right)}^{2}}d\theta }$

(3.24)

Using the integration by part as below

$\displaystyle \int \limits _{0}^{2\pi }{{u_{\theta }}{u_{r,\theta }}d\theta }=\left.{{u_{\theta }}{u_{r}}}\right|_{0}^{2\pi }-\int \limits _{0}^{2\pi }{{u_{r}}{u_{\theta ,\theta }}d\theta =}-\int \limits _{0}^{2\pi }{{u_{r}}{u_{\theta ,\theta }}d\theta }$

(3.25)

to write the integration of last bracket

$\displaystyle \int \limits _{0}^{2\pi }{\left[{{{\left({u_{r}}\right)}^{2}}+{{\left({u_{\theta }}\right)}^{2}}+{u_{r}}{u_{\theta ,\theta }}-{u_{\theta }}{u_{r,\theta }}}\right]d\theta }\leq \int \limits _{0}^{2\pi }{\left[{{{\left({u_{r}}\right)}^{2}}+{\frac {1}{4}}{{\left({u_{\theta ,\theta }}\right)}^{2}}+2{u_{r}}{u_{\theta ,\theta }}}\right]d\theta }=\int \limits _{0}^{2\pi }{\left[{{{\left({{u_{r}}+{u_{\theta ,\theta }}}\right)}^{2}}-{\frac {3}{4}}{{\left({u_{\theta ,\theta }}\right)}^{2}}}\right]d\theta }$

(3.26)

Note that

$\displaystyle {\varepsilon _{\theta \theta }}={\frac {{u_{r}}+{u_{\theta ,\theta }}}{r}}$

(3.27)

one can write the stability condition as

$\displaystyle \int \limits _{A_{1}}{\left\{{2{\mu _{1}}\left[{\left({1-\beta }\right){{\boldsymbol {\varepsilon }}^{'}}{\mathbf {:} }{{\boldsymbol {\varepsilon }}^{'}}+\beta {\mathbf {\omega :\omega } }}\right]+\left[{{\lambda _{1}}+\left({1+\beta }\right){\mu _{1}}}\right]{{\left({tr{\boldsymbol {\varepsilon }}}\right)}^{2}}}\right\}dA}$

(3.28)

$\displaystyle +\int \limits _{0}^{2\pi }{\left\{{ta\left[{2{\mu _{2}}\left({\varepsilon _{rr}^{2}+\varepsilon _{\theta \theta }^{2}+2\varepsilon _{r\theta }^{2}}\right)+{\lambda _{2}}{{\left({{\varepsilon _{rr}}+{\varepsilon _{\theta \theta }}}\right)}^{2}}}\right]+2\beta {\mu _{1}}\left[{{\frac {3}{4}}{{\left({u_{\theta ,\theta }}\right)}^{2}}-{a^{2}}{\varepsilon _{\theta \theta }^{2}}}\right]}\right\}d\theta }>0$

(3.29)

From this, Drugan in concluded that:

$\displaystyle \mu _{1}>0,\quad \lambda _{1}>-(1+\beta )\mu _{1},\quad \mu _{2}>\beta {\frac {a}{t}}\mu _{1}>0,\quad \lambda _{2}>0$

(3.20)

For each inequality, justify where it came from. For example, the requirement that $\displaystyle \mu _{1}>0$ is to force the first integrand in the first term in (3.28), i.e., $\displaystyle {2{\mu _{1}}\left[{\left({1-\beta }\right){{\boldsymbol {\varepsilon }}^{'}}{\mathbf {:} }{{\boldsymbol {\varepsilon }}^{'}}+\beta {\mathbf {\omega :\omega } }}\right]}$ to be positive. Egm6321.f11 15:25, 24 June 2011 (UTC)

Weaker sufficient restrictions for stability are obtained by Drugan, as follows. Ignoring the terms $\displaystyle \epsilon _{r\theta }^{2},\ \left(u_{\theta ,\theta }\right)^{2}$ the integrand in the second integration of (3.20) becomes:

You need to justify why you want to ingnore the terms $\displaystyle \epsilon _{r\theta }^{2},\ \left(u_{\theta ,\theta }\right)^{2}$; the reason is not clear. The reason is because the terms $\displaystyle 4ta\mu _{2}\epsilon _{r\theta }^{2}$ and $\displaystyle {\frac {3}{2}}\beta \mu _{1}\mu _{\theta ,\theta }^{2}$ in (3.29) are already positive; so ignoring these two terms and enforcing the remaining terms to be positive is a stronger requirement. So for now, such stronger requirement led to the inequality (stability condition) $\displaystyle \lambda _{1}>-(1+\beta )\mu _{1}$, which means that $\displaystyle \lambda _{1}$ can be negative. But to conclude that $\displaystyle \nu _{1}$ (Poisson's ratio for material 1) can be negative, you need to look at the relationship between $\displaystyle \lambda _{1}$ and $\displaystyle \nu _{1}$. Now, we have $\displaystyle \nu _{1}={\frac {\lambda _{1}}{2(\lambda _{1}+\mu _{1})}}$ (see Lame' parameters (constants) (wikipedia); verify !), then we need to have the condition $\displaystyle (\lambda _{1}+\mu _{1})>0$, which is the condition for positive definiteness of the elastic moduli tensor $\displaystyle {\boldsymbol {C}}$; verify ! Can you deduce from the general stability condition (2.1.21) for elastic materials that $\displaystyle (\lambda _{1}+\mu _{1})>0$ ? Egm6321.f11 15:25, 24 June 2011 (UTC)

$\displaystyle F=ta[2\mu _{2}(\epsilon _{rr}^{2}+\epsilon _{\theta \theta }^{2})+\lambda _{2}(\epsilon _{rr}+\epsilon _{\theta \theta })^{2}]-2a^{2}\beta \mu _{1}\epsilon _{\theta \theta }^{2}$

(3.21)

We require that this expression is positive or

$\displaystyle G={\frac {F}{ta}}=2\mu _{2}(\epsilon _{rr}^{2}+\epsilon _{\theta \theta }^{2})+\lambda _{2}(\epsilon _{rr}+\epsilon _{\theta \theta })^{2}-2\alpha \epsilon _{\theta \theta }^{2}>0$

(3.22)

where $\displaystyle \alpha =\beta {\frac {a}{t}}\mu _{1}$

(3.22) is equivalent to

$\displaystyle (\lambda _{2}+2\mu _{2})\epsilon _{rr}^{2}+2\lambda _{2}\epsilon _{rr}\epsilon _{\theta \theta }+(\lambda _{2}+2\mu _{2}-2\alpha )\epsilon _{\theta \theta }^{2}>0$

(3.23)

or

$\displaystyle {\begin{matrix}\lambda _{2}+2\mu _{2}>0\\\lambda _{2}^{2}\epsilon _{\theta \theta }^{2}-(\lambda _{2}+2\mu _{2})(\lambda _{2}+2\mu _{2}-2\alpha )\epsilon _{\theta \theta }^{2}<0\end{matrix}}$

(3.24)

Solving this system of two inequalities to get the final condition:

$\displaystyle \mu _{2}>{\frac {\alpha }{2}},\,\,\lambda _{2}>-\mu _{2}{\frac {\mu _{2}-\alpha }{\mu _{2}-{\frac {\alpha }{2}}}}=-\mu _{2}{\frac {1-\beta {\frac {a}{t}}{\frac {\mu _{1}}{\mu _{2}}}}{1-\beta {\frac {a}{t}}{\frac {\mu _{1}}{\mu _{2}}}}}$

(3.25)

This condition (3.25) is "better" than the condition in (3.20) since it provides a larger stability domain (and hence more choices of material properties) than that from (3.20). Note that if $\displaystyle \mu _{1}>\alpha$, and since $\displaystyle \mu _{1}>0$, then $\displaystyle -\mu _{1}{\frac {\mu _{1}-\alpha }{\mu _{1}-{\frac {\alpha }{2}}}}<0$ and thus $\displaystyle \lambda _{1}$ can be negative. Egm6321.f11 15:25, 24 June 2011 (UTC)