Let $\Omega$ be a region in $\mathbb{R}^2$ and let its boundary $\Gamma = \overline{\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4}$ be composed of non overlapping subregions of $\Gamma$. Let $n$ be the unit outward normal vector to $\Gamma$ such that $s$ and $n$ form a right-hand rule basis. Consider the following BVP in classical linear elastostatics
find $u_i : \overline{\Omega}\rightarrow\mathbb{R}$ such that
σij,j+fi=0ui=qiσijnj=hi
where $\sigma{ij} = c{ijkl}u_{(k,l)}$. Establish a weak formulation for this problem in which all “$q$-type” boundary conditions are essential and all “$h$-type ” boundary conditions are natural. State all requirements on the spaces $\delta$ and $\mathcal{V}$. Hint $w_i = w_n n_i+ w_s s_i$.
The boundary $\Gamma_{h_i}$ is the union of the 4 non overlapping boundaries per $\Gamma = \overline{\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4}$. Each correspond to one of the given kinds of boundary conditions, which we are asked to generalize. Investigation of the boundary contour integral requires one integral.
Function Space Assumptions
δ={u∣u∈H1,ui=gi on Γgi}V={w∣w∈H1,wi=0 on Γgi}
The usual decomposition of these function spaces such that our finite spaces are $\delta^h \in \delta \text{ and } \mathcal{V}^h \in \mathcal{V}$.
In practice it is often useful to generalize the constitutive equation of classical elasticity to the form
σij=cijkl(ϵkl−ϵkl0)+σij0
Where $\epsilon{ij}^0$ and $\sigma{ij}^0$
are the initial strain and initial stress both given functions of $x$. The initial strain term may be used to represent thermal expansion effects by way of
ϵkl0=−θckl
where $\theta$ is the temperature and the $c_{kl}$’s are the thermal expansion coefficients (both given functions). Clearly [const eqn] will in no way change the stiffness matrix . However there will be additional contributions to $f_p^e$ Generalize the definition of $f_p^e$ to account for these additional terms.
The mesh and boundary conditions given below are for a 2-D elastostatics problem. Following the method given in the class handout, determine the ID matrix, IEN vectors and LM matrix. Also indicate the locations in the stiffness matrix, or force vector the “contributions” for the terms in the element stiffness matrix for element 7 will go. Essential BC information (be sure to account for it): The vertical edge (nodes 1,4,7,8 are on it) can not move in the in either direction The horizontal edge (nodes 15, 3,13 on it) has a prescribed vertical displacement of 0.05 units. Note – Nodes are marked with a black dot (there are corner nodes and three edge nodes). Element numbers have the circles around them.
Let $n$ be $null$, $f$ be a degree of freedom, and $g$ be a nonzero essential boundary condition. As $n_\text{sd} =2$, the first entry corresponds to the $x$ direction and the second with $y$.