branch: master commit 4164a973b3cbab2e56808a7971d8aff5248d04a8 Author: Yves Renard <yves.ren...@insa-lyon.fr> AuthorDate: Fri Nov 20 08:34:37 2020 +0100
fix formula typos --- doc/sphinx/source/userdoc/model_nonlinear_elasticity.rst | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/doc/sphinx/source/userdoc/model_nonlinear_elasticity.rst b/doc/sphinx/source/userdoc/model_nonlinear_elasticity.rst index af9264a..d4eeb9d 100644 --- a/doc/sphinx/source/userdoc/model_nonlinear_elasticity.rst +++ b/doc/sphinx/source/userdoc/model_nonlinear_elasticity.rst @@ -113,7 +113,7 @@ The stress in the reference configuration can be describe by the second Piola-Ki .. math:: - {\hat{\hat{\sigma}}} &= \frac{\partial}{\partial E} {W}(E) = 2\frac{\partial}{\partial C} {W}(C) + {\hat{\hat{\sigma}}} = \frac{\partial}{\partial E} {W}(E) = 2\frac{\partial}{\partial C} {W}(C) where :math:`{W}` is the density of strain energy of the material. The total strain energy is given by @@ -222,12 +222,12 @@ Incompressible material. .. math:: {d_1} = 0 - \intertext{with the additional constraint:} + \mbox{ with the additional constraint: } i_3( C) = 1 The incompressibility constraint :math:`i_3( C) = 1` is handled with a Lagrange multiplier :math:`p` (the pressure) -constraint: :math:`\sigma = -pI \rm I\hspace{-0.15em}Rightarrow {\hat{\hat{\sigma}}} = -p\nabla\Phi\nabla\Phi^{-T}\det\nabla\Phi` +constraint: :math:`\sigma = -pI \Rightarrow {\hat{\hat{\sigma}}} = -p\nabla\Phi\nabla\Phi^{-T}\det\nabla\Phi` .. math:: @@ -248,7 +248,7 @@ constraint: :math:`\sigma = -pI \rm I\hspace{-0.15em}Rightarrow {\hat{\hat{\sigm .. math:: - {W} &= a\; i_1(C) + (\frac{\mu}{2} - a)i_2(C) + (\frac{\lambda}{4} - \frac{\mu}{2} + a)i_3(C) - (\frac{\mu}{2}+\frac{\lambda}{4})\log \det(C) + {W} = a\; i_1(C) + (\frac{\mu}{2} - a)i_2(C) + (\frac{\lambda}{4} - \frac{\mu}{2} + a)i_3(C) - (\frac{\mu}{2}+\frac{\lambda}{4})\log \det(C) with :math:`\lambda, \mu` the Lame coefficients and :math:`\max(0,\frac{\mu}{2}-\frac{\lambda}{4})<a<\frac{\mu}{2}` (see [ciarlet1988]_). @@ -258,14 +258,14 @@ with :math:`\lambda, \mu` the Lame coefficients and :math:`\max(0,\frac{\mu}{2} .. math:: - {W} &= (ai_1(C) + bi_3(C)^{1/2} + c\frac{\i_2(C)}{\i_3(C)} + d)^n + {W} = (a i_1(C) + b i_3(C)^{1/2} + c\frac{i_2(C)}{i_3(C)} + d)^n Since :math:`\frac{\partial}{\partial C} {W}(C) = \displaystyle\sum_{j}\frac{\partial W}{\partial i_j(C)} \frac{\partial i_j(C)}{\partial C}`, and :math:`\frac{\partial^2}{\partial C^2} {W}(C) = \displaystyle\sum_{j} \displaystyle\sum_{k} \frac{\partial^2 W}{\partial i_j(C) \partial i_k(C)} \frac{\partial i_k(C)}{\partial C} \otimes \frac{\partial i_j(C)}{\partial C} + \displaystyle\sum_{j} \frac{\partial W}{\partial i_j(C)} \frac{\partial^2 i_j(C)}{\partial C^2}` we must compute the der [...] .. math:: \begin{array}{l} \frac{\partial W}{\partial i_1(C)} = naZ^{n-1} - ~~~~\mbox{with } Z = (ai_1(C) + bi_3(C)^{1/2} + c\frac{\i_2(C)}{\i_3(C)} + d)\\ + ~~~~\mbox{with } Z = (a i_1(C) + b i_3(C)^{1/2} + c\frac{i_2(C)}{i_3(C)} + d)\\ \frac{\partial W}{\partial i_2(C)} = n\frac{c}{i_3(C)}Z^{n-1}\\ \frac{\partial W}{\partial i_3(C)} = n(\frac{b}{2i_3(C)^{1/2}}-\frac{ci_2(C)}{i_3(C)^2})Z^{n-1}\\ \frac{\partial W^2}{\partial^2 i_1(C)} = n(n-1)A^2Z^{n-2}\\