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| {{Continuum mechanics|cTopic=[[Solid mechanics]]}}
| | The recent [http://answers.yahoo.com/search/search_result?p=optimism+asserted&submit-go=Search+Y!+Answers optimism asserted] itself again Friday supported by U.S.corporate earnings and some blind faith the Europeans can fix their debt crisis. At least they appear to be working on it with some harmony from recent rhetoric.<br><br>A place where respected investment gurus, TV talking heads and even celebrities like Shaquille O' Neil sing the praises of worthless penny stocks, while promising you' ll get rich thanks to returns of 1,000% or more virtually overnight.<br><br><br><br>Next, we had a group of questions relating to virtual or simulated stock trading? Apparently, new traders want to test the waters before making a capital investment. Not a bad idea.<br><br>This allows you to have a cushion if you lose a job, unemployment costs, or even damage from a disaster which might not be covered by insurance until you get your affairs in order.<br><br>The going gets tough once again. How ever does one learn the pros and cons of which products to invest in? True, the title of this article involves investing in stocks but let's expand on that. Any type of investment will be accompanied by a steep learning curve. The best way to follow that curve is not by laying out your hard-earned money at random and hope one of your investments hits pay dirt. You need an experienced financial guru to show you the ropes.<br><br>A lot of commentators talk about how gold is near an all-time high and that stocks have fallen 50%, making them cheap again. However from a [http://Www.long-term.org/ long-term] perspective, gold and stocks are nowhere near their normal relationship.<br><br>Why is trading the eMini better than trading stocks? There are many reasons, in my opinion, but I will discuss one of them here, and other reasons in future articles. I prefer eMinis to stocks because they avoid certain types of risks. In economics the term exogenous is used to refer to an event that occurs "from outside" the system, model, or idea you are considering. It usually is an unexpected event that creates a shock to the system. For some traders, exogenous shocks can result in a windfall of profits, but for most traders, they result in [http://news.goldgrey.org/precious-metals/ rhodium] losses in their brokerage accounts-which leaves them shocked.<br><br>There's a fairly large subset of humanity who can never be successful at stockmarket investment but not because they're not clever enough. The trouble is that when the going gets tough - and you can guarantee that at some point it will - most people can't take the heat.<br><br>Turn on your personal computer and get the membership of various stock trading communities. Within a few months you will come to know the benefits of these communities. At the initial months, you should focus at the research of online stock trading and after you have got satisfied with the gained knowledge you should go forward for further investments. Never take the help of brokers or traders .Instead of it, you should trust your stock trading community. |
| The ''' [[Alan N. Gent|Gent]]''' [[hyperelastic material]] model <ref name=Gent/> is a phenomenological model of [[rubber elasticity]] that is based on the concept of limiting chain extensibility. In this model, the [[strain energy density function]] is designed such that it has a [[mathematical singularity|singularity]] when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value <math>I_m</math>. | |
| | |
| The strain energy density function for the Gent model is <ref name=Gent>Gent, A.N., 1996, '' A new constitutive relation for rubber'', Rubber Chemistry Tech., 69, pp. 59-61.</ref>
| |
| :<math>
| |
| W = -\cfrac{\mu J_m}{2} \ln\left(1 - \cfrac{I_1-3}{J_m}\right)
| |
| </math>
| |
| where <math>\mu</math> is the [[shear modulus]] and <math>J_m = I_m -3</math>.
| |
| | |
| In the limit where <math>I_m \rightarrow \infty</math>, the Gent model reduces to the [[Neo-Hookean solid]] model. This can be seen by expressing the Gent model in the form
| |
| :<math>
| |
| W = \cfrac{\mu}{2x}\ln\left[1 - (I_1-3)x\right] ~;~~ x := \cfrac{1}{J_m}
| |
| </math>
| |
| A [[Taylor series expansion]] of <math>\ln\left[1 - (I_1-3)x\right]</math> around <math>x = 0</math> and taking the limit as <math>x\rightarrow 0</math> leads to | |
| :<math>
| |
| W = \cfrac{\mu}{2} (I_1-3)
| |
| </math>
| |
| which is the expression for the strain energy density of a Neo-Hookean solid.
| |
| | |
| Several '''compressible''' versions of the Gent model have been designed. One such model has the form<ref>Mac Donald, B. J., 2007, '''Practical stress analysis with finite elements''', Glasnevin, Ireland.</ref>
| |
| :<math>
| |
| W = -\cfrac{\mu J_m}{2} \ln\left(1 - \cfrac{I_1-3}{J_m}\right) + \cfrac{\kappa}{2}\left(\cfrac{J^2-1}{2} - \ln J\right)^4
| |
| </math>
| |
| where <math>J = \det(\boldsymbol{F})</math>, <math>\kappa</math> is the [[bulk modulus]], and <math>\boldsymbol{F}</math> is the [[deformation gradient]].
| |
| | |
| == Consistency condition ==
| |
| We may alternatively express the Gent model in the form
| |
| :<math>
| |
| W = C_0 \ln\left(1 - \cfrac{I_1-3}{J_m}\right)
| |
| </math> | |
| For the model to be consistent with [[linear elasticity]], the [[Hyperelastic_material#Consistency_conditions_for_incompressible_I1_based_rubber_materials|following condition]] has to be satisfied:
| |
| :<math>
| |
| 2\cfrac{\partial W}{\partial I_1}(3) = \mu
| |
| </math>
| |
| where <math>\mu</math> is the [[shear modulus]] of the material.
| |
| Now, at <math>I_1 = 3 (\lambda_i = \lambda_j = 1)</math>,
| |
| :<math>
| |
| \cfrac{\partial W}{\partial I_1} = -\cfrac{C_0}{J_m}
| |
| </math>
| |
| Therefore, the consistency condition for the Gent model is
| |
| :<math>
| |
| -\cfrac{2C_0}{J_m} = \mu\, \qquad \implies \qquad C_0 = -\cfrac{\mu J_m}{2}
| |
| </math>
| |
| The Gent model assumes that <math>J_m \gg 1</math> | |
| | |
| == Stress-deformation relations ==
| |
| The Cauchy stress for the incompressible Gent model is given by
| |
| :<math>
| |
| \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} +
| |
| 2~\cfrac{\partial W}{\partial I_1}~\boldsymbol{B}
| |
| = -p~\boldsymbol{\mathit{1}} + \cfrac{\mu J_m}{J_m - I_1 + 3}~\boldsymbol{B}
| |
| </math>
| |
| | |
| === Uniaxial extension ===
| |
| [[Image:Hyperelastic.svg|thumb|350px|right|Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models.]]
| |
| For uniaxial extension in the <math>\mathbf{n}_1</math>-direction, the [[finite strain theory|principal stretches]] are <math>\lambda_1 = \lambda,~ \lambda_2=\lambda_3</math>. From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>. Hence <math>\lambda_2^2=\lambda_3^2=1/\lambda</math>.
| |
| Therefore,
| |
| :<math> | |
| I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{2}{\lambda} ~.
| |
| </math>
| |
| The [[finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
| |
| :<math>
| |
| \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{1}{\lambda}~(\mathbf{n}_2\otimes\mathbf{n}_2+\mathbf{n}_3\otimes\mathbf{n}_3) ~.
| |
| </math>
| |
| If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
| |
| :<math>
| |
| \sigma_{11} = -p + \cfrac{\lambda^2\mu J_m}{J_m - I_1 + 3} ~;~~
| |
| \sigma_{22} = -p + \cfrac{\mu J_m}{\lambda(J_m - I_1 + 3)} = \sigma_{33} ~.
| |
| </math>
| |
| If <math>\sigma_{22} = \sigma_{33} = 0</math>, we have
| |
| :<math>
| |
| p = \cfrac{\mu J_m}{\lambda(J_m - I_1 + 3)}~.
| |
| </math>
| |
| Therefore,
| |
| :<math>
| |
| \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
| |
| </math>
| |
| The [[stress (physics)|engineering strain]] is <math>\lambda-1\,</math>. The [[stress (physics)|engineering stress]] is
| |
| :<math>
| |
| T_{11} = \sigma_{11}/\lambda =
| |
| \left(\lambda - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
| |
| </math>
| |
| | |
| === Equibiaxial extension ===
| |
| For equibiaxial extension in the <math>\mathbf{n}_1</math> and <math>\mathbf{n}_2</math> directions, the [[finite strain theory|principal stretches]] are <math>\lambda_1 = \lambda_2 = \lambda\,</math>. From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>. Hence <math>\lambda_3=1/\lambda^2\,</math>.
| |
| Therefore,
| |
| :<math>
| |
| I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = 2~\lambda^2 + \cfrac{1}{\lambda^4} ~.
| |
| </math>
| |
| The [[finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
| |
| :<math>
| |
| \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \lambda^2~\mathbf{n}_2\otimes\mathbf{n}_2+ \cfrac{1}{\lambda^4}~\mathbf{n}_3\otimes\mathbf{n}_3 ~.
| |
| </math>
| |
| If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
| |
| :<math>
| |
| \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda^4}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) = \sigma_{22} ~.
| |
| </math>
| |
| The [[stress (physics)|engineering strain]] is <math>\lambda-1\,</math>. The [[stress (physics)|engineering stress]] is
| |
| :<math>
| |
| T_{11} = \cfrac{\sigma_{11}}{\lambda} =
| |
| \left(\lambda - \cfrac{1}{\lambda^5}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) = T_{22}~.
| |
| </math>
| |
| | |
| === Planar extension ===
| |
| Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the <math>\mathbf{n}_1</math> directions with the <math>\mathbf{n}_3</math> direction constrained, the [[finite strain theory|principal stretches]] are <math>\lambda_1=\lambda, ~\lambda_3=1</math>. From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>. Hence <math>\lambda_2=1/\lambda\,</math>.
| |
| Therefore,
| |
| :<math>
| |
| I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{1}{\lambda^2} + 1 ~.
| |
| </math>
| |
| The [[finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
| |
| :<math>
| |
| \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{1}{\lambda^2}~\mathbf{n}_2\otimes\mathbf{n}_2+ \mathbf{n}_3\otimes\mathbf{n}_3 ~.
| |
| </math>
| |
| If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
| |
| :<math>
| |
| \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) ~;~~ \sigma_{22} = 0 ~;~~ \sigma_{33} = \left(1 - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
| |
| </math>
| |
| The [[stress (physics)|engineering strain]] is <math>\lambda-1\,</math>. The [[stress (physics)|engineering stress]] is
| |
| :<math>
| |
| T_{11} = \cfrac{\sigma_{11}}{\lambda} =
| |
| \left(\lambda - \cfrac{1}{\lambda^3}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
| |
| </math>
| |
| | |
| === Simple shear ===
| |
| The deformation gradient for a [[simple shear]] deformation has the form<ref name=Ogden>Ogden, R. W., 1984, '''Non-linear elastic deformations''', Dover.</ref>
| |
| :<math>
| |
| \boldsymbol{F} = \boldsymbol{1} + \gamma~\mathbf{e}_1\otimes\mathbf{e}_2
| |
| </math>
| |
| where <math>\mathbf{e}_1,\mathbf{e}_2</math> are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
| |
| :<math>
| |
| \gamma = \lambda - \cfrac{1}{\lambda} ~;~~ \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac{1}{\lambda} ~;~~ \lambda_3 = 1
| |
| </math>
| |
| In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
| |
| :<math>
| |
| \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ~;~~
| |
| \boldsymbol{B} = \boldsymbol{F}\cdot\boldsymbol{F}^T = \begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
| |
| </math>
| |
| Therefore,
| |
| :<math>
| |
| I_1 = \mathrm{tr}(\boldsymbol{B}) = 3 + \gamma^2
| |
| </math>
| |
| and the Cauchy stress is given by
| |
| :<math>
| |
| \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + \cfrac{\mu J_m}{J_m - \gamma^2}~\boldsymbol{B}
| |
| </math>
| |
| In matrix form,
| |
| :<math>
| |
| \boldsymbol{\sigma} = \begin{bmatrix} -p +\cfrac{\mu J_m (1+\gamma^2)}{J_m - \gamma^2} & \cfrac{\mu J_m \gamma}{J_m - \gamma^2} & 0 \\ \cfrac{\mu J_m \gamma}{J_m - \gamma^2} & -p + \cfrac{\mu J_m}{J_m - \gamma^2} & 0 \\ 0 & 0 & -p + \cfrac{\mu J_m}{J_m - \gamma^2}
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| == See also ==
| |
| * [[Hyperelastic material]]
| |
| * [[Strain energy density function]]
| |
| * [[Mooney-Rivlin solid]]
| |
| * [[Finite strain theory]]
| |
| * [[Stress measures]]
| |
| | |
| [[Category:Continuum mechanics]]
| |
| [[Category:Elasticity (physics)]]
| |
| [[Category:Non-Newtonian fluids]]
| |
| [[Category:Rubber properties]]
| |
| [[Category:Solid mechanics]]
| |
The recent optimism asserted itself again Friday supported by U.S.corporate earnings and some blind faith the Europeans can fix their debt crisis. At least they appear to be working on it with some harmony from recent rhetoric.
A place where respected investment gurus, TV talking heads and even celebrities like Shaquille O' Neil sing the praises of worthless penny stocks, while promising you' ll get rich thanks to returns of 1,000% or more virtually overnight.
Next, we had a group of questions relating to virtual or simulated stock trading? Apparently, new traders want to test the waters before making a capital investment. Not a bad idea.
This allows you to have a cushion if you lose a job, unemployment costs, or even damage from a disaster which might not be covered by insurance until you get your affairs in order.
The going gets tough once again. How ever does one learn the pros and cons of which products to invest in? True, the title of this article involves investing in stocks but let's expand on that. Any type of investment will be accompanied by a steep learning curve. The best way to follow that curve is not by laying out your hard-earned money at random and hope one of your investments hits pay dirt. You need an experienced financial guru to show you the ropes.
A lot of commentators talk about how gold is near an all-time high and that stocks have fallen 50%, making them cheap again. However from a long-term perspective, gold and stocks are nowhere near their normal relationship.
Why is trading the eMini better than trading stocks? There are many reasons, in my opinion, but I will discuss one of them here, and other reasons in future articles. I prefer eMinis to stocks because they avoid certain types of risks. In economics the term exogenous is used to refer to an event that occurs "from outside" the system, model, or idea you are considering. It usually is an unexpected event that creates a shock to the system. For some traders, exogenous shocks can result in a windfall of profits, but for most traders, they result in rhodium losses in their brokerage accounts-which leaves them shocked.
There's a fairly large subset of humanity who can never be successful at stockmarket investment but not because they're not clever enough. The trouble is that when the going gets tough - and you can guarantee that at some point it will - most people can't take the heat.
Turn on your personal computer and get the membership of various stock trading communities. Within a few months you will come to know the benefits of these communities. At the initial months, you should focus at the research of online stock trading and after you have got satisfied with the gained knowledge you should go forward for further investments. Never take the help of brokers or traders .Instead of it, you should trust your stock trading community.