Category: Uncategorized

CBT by variational approach
Consider a straight beam under lateral loading q(x) and axial force P. The classical beam theory (CBT) makes kinematic assumption as \begin{align*} u&=u_0(x)z\frac{dw_0}{dx}\\ v&=0\\ w&=w_0(x) \end{align*} The strain therefore is \begin{align*} &\epsilon_x=\frac{du_0}{dx}z\frac{d^2w_0}{dx^2},\epsilon_y=\epsilon_z=0\\ &\gamma_{xy}=\gamma_{xz}=\gamma_{yz}=0 \end{align*} Let’s follow virtual work principle \delta U=\delta W where internal virtual work is \delta U=\int_V\sigma_x\delta\epsilon_xdV Let’s expand it \begin{align*} \delta U&=\int_0^L\int_0^b\int_{h/2}^{h/2}E\left(\frac{du_0}{dx}z\frac{d^2w_0}{dx^2}\right)\left(\frac{d\delta…

Hyperelasticity: uniaxial tension of Gent material
An incompressible Gent model is written as \psi=\frac{\mu}2J_m\ln\left(1\frac{I_13}{J_m}\right) Given a Cartesian coordinate, a uniaxial tension is applied along x_1 direction. The deformation gradient is assumed as \bold{F}=\begin{bmatrix} \lambda_1&&\\ &\lambda_0&\\ &&\lambda_0 \end{bmatrix} Where stretch along x_2,x_3 should be same, \lambda_2=\lambda_3=\lambda_0, due to isotropy. We can therefore calculate C matrix and invariant \bold{C}=\bold{F}\bold{F}^T=\begin{bmatrix}\lambda^2_1&&\\&\lambda^2_0&\\&&\lambda^2_0\end{bmatrix},\ I_1=tr(\bold{C})=\lambda^2_1+2\lambda^2_0 Stress is calculated…

Geometry in mechanics: Linear space
Linear spacce In short, a linear space (also called vector space) is closed under ‘addtion’ and ‘scalar multiplication’ operation. space To define a linear space (also called vector space), we will firstly need a number field F and a nonempty set \bold{V}. The number field can be integer, real number or complex number, etc. The…

FEM: Material tangent of neoHookean material
Let’s calculate material tangent of hyperelastic material, taking neoHookean for instance. The elastic energy of neoHookean material is written as \psi^e=\frac{\mu}2tr(\bold{C})\mu\ln J A penalty method is applied to constrain volume unchanged \psi^b=\frac{\kappa}2(J^22\ln J1) The total free energy density is therefore written into \psi=\psi^e+\psi^b The stress is obtained by derivative P_{iA}=\frac{\partial\psi}{\partial F_{iA}}=\frac{\partial\psi^e}{\partial F_{iA}}+\frac{\partial\psi^b}{\partial F_{iA}} The elastic…

FEM: Fbar in total lagrangian shceme
Fbar method is a element technique used in linear elements to alleviate volumetric locking. This method replaces F of each integration point with Fbar. Index notation Displacement interpolation u_i=N_{iK}d_k Deformation gradient is calculated as F_{iA}=\delta_{iA}+N_{iK,A}d_K=\delta_{iA}+B_{iKA}d_K where d_K is nodal displacement Total lagrangian shceme without traction and body force is \delta U=\int P_{iA}\delta F_{iA}dV and nodal…