In this module, we will present the strong and weak formulations for the equations of linear elasticity theory, which are very commonly used in calculations using the finite element method for stresses in various structures.
Consider, for example, a three-dimensional platform supported by five supports, standing on the OXY plane, shown in Fig. 1. This platform has its own weight and certain stresses appear on it due to the influence of the gravitational field of the earth.
For a more convenient formulation of the partial differential equations describing the displacement of the platform, we will now use the notation \( (x_1,x_2,x_3) \) at any point of the platform.
The solution sought this time will be the vector field of displacements
\( \Omega \ni (x_1,x_2,x_3) \rightarrow (u_1(x_1,x_2,x_3),u_2(x_1,x_2,x_3),u_3(x_1,x_2,x_3)) \)
This means in particular that the displacement of the platform points under its weight in the direction of the axis \( z \), are equal \( u_3(x_1,x_2,x_3) \) in point \( (x_1,x_2,x_3) \). Of course, the gravitational field works only in the direction of the Z axis, but the platform points are "glued" to each other and if the platform point moves in the Z direction due to the weight of the platform, it is often associated with a point shift in the X and Y directions, which is described by the components \( u_1(x_1,x_2,x_3) \) and \( u_2(x_1,x_2,x_3) \). Of course, it is not easy to predict how these points will deform, and in particular it requires solving complicated equations of the linear theory of elasticity.
To simplify the notation and due to the use of traditional notation, we introduce a designation for partial derivatives of individual coordinates of the displacement vector
\( u_{i,j}=\frac{\partial u_i}{\partial x_j } \)
which represents the derivative in the \( j \)-th direction of the \( i \)-th component of the vector field.
In order to derive the equations of the linear theory of elasticity, a so-called strain rate tensor is introduced
\( \epsilon_{ij}=\frac{\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}}{2}=\frac{u_{i,j}+u_{j,i } }{2 } \)
and stress tensor
\( \sigma_{ij}=c_{ijkl}\epsilon_{kl} \)
We use the so-called Einstein summation convention, in which duplicating the indices in the formula means that we sum over these indices, i.e. in the equation of elasticity
\( \sigma_{ij}=c_{ijkl}\epsilon_{kl } \)
repetition of indexes \( k,l \) means summotion
\( \sigma_{ij}=\sum_{k=1,...,3;l=1,...,3} c_{ijkl}\epsilon_{kl } \).
We have a four-dimensional matrix of coefficients here \( c_{ijkl},i=1,...,3;j=1,...,3,k=1,...,3;l=1,..,3 \). The definition of these 3 * 3 * 3 * 3 numbers clearly defines the material from which the structure I am analyzing is made of (the platform in our example). Fortunately, most of these factors will be zero.
The so-called strong formulation of the linear theory of elasticity problem is formulated as follows
\( \sigma_{ij,j}+f_i=0 \) for i=1,2,3,inside \( \Omega \). The term \( f_i \) stands for 3 coordinates of the force vector acting on our structure, for example it will be the force of gravity \( f=(g_1(x_1,x_2,x_3),g_2(x_1,x_2,x_3),g_3(x_1,x_2,x_3)) \)
We introduce the following convention here. Indices repeated after the decimal point \( ij,j \) denote the computation of the partial derivative after the indexed variable.
Term \( \sigma_{ij,j}=\frac{\partial \sigma_{ij} }{\partial x_j } \) thus denotes the sum of partial derivatives of the stress tensor. Moreover, we have three equations defacto (which makes sense since we are looking for the three coordinates of the displacement vector).
\( \sum_{j=1,2,3}\frac{\partial \sigma_{1j}}{\partial x_j}+f_1=0 \\ \sum_{j=1,2,3}\frac{\partial \sigma_{2j}}{\partial x_j }+f_2=0 \\ \sum_{j=1,2,3}\frac{\partial \sigma_{3j}}{\partial x_j }+f_3=0 \)
that is, taking into account the definition of the stress tensor
\( \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{1jkl}\epsilon_{kl})}{\partial x_j}+f_1=0 \\ \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{2jkl}\epsilon_{kl})}{\partial x_j}+f_2=0 \\ \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{3jkl}\epsilon_{kl})}{\partial x_j }+f_3=0 \)
that is, taking into account the definition of the strain velocity tensor
\( \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{1jkl}(\frac{u_{k,l}+u_{l,k}}{2}))}{\partial x_j }+f_1=0 \\ \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{2jkl}(\frac{u_{k,l}+u_{l,k}}{2}))}{\partial x_j }+f_2=0 \\ \sum_{j=1,2,3}\frac{\partial (\sum_{k=1,...,3;l=1,...,3} c_{3jkl}\frac{u_{k,l}+u_{l,k}}{2})}{\partial x_j }+f_3=0 \)
Our unknown here is the displacement vector field: We are looking at
\( \Omega \ni (x_1,x_2,x_3) \rightarrow (u_1(x_1,x_2,x_3),u_2(x_1,x_2,x_3),u_3(x_1,x_2,x_3)) \)
In order to solve the equations of the linear theory of elasticity on a given structure, it is necessary to define the boundary conditions, i.e. to define what is happening at the edge of the structure. On part of the boundary, we can introduce a Dirichlet condition, for example specifying that the base of the structure is attached to the ground, and therefore the displacement of the base of the structure is zero
\( u_1(x_1,x_2,x_3)=0, u_2(x_1,x_2,x_3)=0, u_3(x_1,x_2,x_3)=0, \textrm{ dla } (x_1,x_2,x_3)\in \Gamma_D \)
It is also possible to define the Neumann boundary condition
\( \sigma_{ij}n_j = h_i \textrm{ dla } (x_1,x_2,x_3)\in \Gamma_N \)
where \( n(x_1,x_2,x_3)=(n_1(x_1,x_2,x_3),n_2(x_1,x_2,x_3),n_3(x_1,x_2,x_3)) \) which means versor perpendicular to the boundary at point \( (x_1,x_2,x_3) \), and \( h=(h_1(x_1,x_2,x_3),h_2(x_1,x_2,x_3),h_3(x_1,x_2,x_3)) \) denotes a given function on the Neumann edge \( \Gamma_N \), that is, taking into account Einstein's summation conventions
\( \sigma_{11}n_2+\sigma_{12}n_2+\sigma_{13}n_3 = h_1 \\ \sigma_{21}n_2+\sigma_{22}n_2+\sigma_{23}n_3 = h_2 \\ \sigma_{31}n_2+\sigma_{32}n_2+\sigma_{33}n_3 = h_3 \)
and also taking into account the definition of the stress tensor
\( (\sum_{k=1,...,3;l=1,...,3} c_{11kl}\epsilon_{kl})n_2+ (\sum_{k=1,...,3;l=1,...,3} c_{12kl}\epsilon_{kl})n_2+ (\sum_{k=1,...,3;l=1,...,3} c_{13kl}\epsilon_{kl})n_3 = h_1 \\ (\sum_{k=1,...,3;l=1,...,3} c_{21kl}\epsilon_{kl})n_2+(\sum_{k=1,...,3;l=1,...,3} c_{22kl}\epsilon_{kl})n_2+ (\sum_{k=1,...,3;l=1,...,3} c_{23kl}\epsilon_{kl})n_3 = h_2 \\ (\sum_{k=1,...,3;l=1,...,3} c_{31kl}\epsilon_{kl})n_2+ (\sum_{k=1,...,3;l=1,...,3} c_{32kl}\epsilon_{kl})n_2+ (\sum_{k=1,...,3;l=1,...,3} c_{33kl}\epsilon_{kl })n_3 = h_3 \).
The Neumann condition defines the forces in the equations of the theory of elasticity.
We give a weak (also called variational) formulation that can be implemented using the finite element method based on the book [1].
The weak (variational) formulation given below is valid for a certain broad class of materials for which the mechanical properties can be described using two parameters, Young's modulus E and the Poisson's ratio \( E \) and the Poisson's ratio \( \mu \).
For a given force field
\( \Omega \ni (x_1,x_2,x_3) \rightarrow f=(g_1(x_1,x_2,x_3),g_2(x_1,x_2,x_3),g_3(x_1,x_2,x_3)) \)
and a given Neumann boundary condition function
\( \Omega \ni (x_1,x_2,x_3) \rightarrow h=(h_1(x_1,x_2,x_3),h_2(x_1,x_2,x_3),h_3(x_1,x_2,x_3)) \)
and a given tensor (four-dimensional matrix) of material coefficients \( c_{ijkl} \) calculate the displacement area of the structure material
\( \Omega \ni (x_1,x_2,x_3) \rightarrow (u_1(x_1,x_2,x_3),u_2(x_1,x_2,x_3),u_3(x_1,x_2,x_3)) \)
satisfying the equations of linear elasticity
\( a(w,u)=(w,f)+(w,h)_{\Gamma} \)
for arbitrary test functions \( w=(w_1(x_1,x_2,x_3),w_2(x_1,x_2,x_3),w_3(x_1,x_2,x_3)) \), where
\( (w,f)=\int_{\Omega}w_1(x_1,x_2,x_3)f_1(x_1,x_2,x_3)dx+\int_{\Omega}w_2(x_1,x_2,x_3)f_2(x_1,x_2,x_3)dy+\int_{\Omega}w_3(x_1,x_2,x_3)f_3(x_1,x_2,x_3)dz \\ (w,h)=\int_{\Gamma}w_1(x_1,x_2,x_3)h_1(x_1,x_2,x_3)dS+\int_{\Gamma}w_2(x_1,x_2,x_3)h_2(x_1,x_2,x_3)dS+\int_{\Gamma}w_3(x_1,x_2,x_3)f_3(x_1,x_2,x_3)dS \\a(w,u)=\int_{\Omega}\epsilon(w)^T D \epsilon(u) dxdydz \)
where
\( \epsilon(u) = \begin{bmatrix} u_{1,1} \\ u_{2,2} \\ u_{3,3} \\ u_{2,3}+u_{3,2} \\ u_{1,3}+u_{3,1} \\ u_{1,2}+u_{2,1} \\ \end{bmatrix}, \epsilon(w) = \begin{bmatrix} w_{1,1} \\ w_{2,2} \\ w_{3,3} \\ w_{2,3}+w_{3,2} \\ w_{1,3}+w_{3,1} \\ w_{1,2}+w_{2,1} \\ \end{bmatrix} \)
and
\( D = \frac{E}{(1+2\mu)(1-2\mu) } \begin{bmatrix} 1-\mu & \mu & \mu & 0 & 0 & 0 \\ \mu & 1-\mu & \mu & 0 & 0 & 0 \\ \mu & \mu & 1-\mu & 0 & 0 & 0 \\ 0 & 0 & 0& \frac{1-2\mu}{2} & 0 & 0 \\ 0 & 0 & 0& 0 & \frac{1-2\mu}{2} & 0 \\ 0 & 0 & 0& 0 & 0& \frac{1-2\mu}{2} \\ \end{bmatrix} \)
is derived with the assumption that the general tensor of material properties \( c_{ijkl} \) can be described by two Young's modulus parameters \( E \) and Poissona coefficient \( \mu \). The given variation formulation is relatively easy to implement. The isogeometric finite element method or the classical finite element method can be used for this purpose.
Consider the following example.

Example 1: Platform deformation

Let us assume that our area \( \Omega \) is a platform with dimensions of 16 by 16 blocks, supported by five supports placed under blocks (1,1), (16,1), (1,16), (16,16) and (8,8), as shown in Fig. 1.

Figure 1: Platform on five supports.

We assume a zero Dirichlet condition at the base of the platform (i.e., we assume that the legs are attached to a flat, stable ground), while on the rest of the edge we do not assume any boundary condition. Such an assumption is sufficient because fixing the platform at the base of the legs gives us the possibility of obtaining an unambiguous solution (calculation of deformations resulting from the weight of the platform). We assume gravitational force \( f=(0,0,-1) \) and \( h=0 \) (no Neumann b.c.)
We have
\( a(w,u)=(w,f)+(w,h)_{\Gamma} \\ a(w,u)=\int_{\Omega}\epsilon(w)^T D \epsilon(u) dxdydz \\(w,f)=\int_{\Omega}w_1(x_1,x_2,x_3)f_1(x_1,x_2,x_3)dx +\int_{\Omega}w_2(x_1,x_2,x_3)f_2(x_1,x_2,x_3)dy + \\ + \int_{\Omega}w_3(x_1,x_2,x_3)f_3(x_1,x_2,x_3)dz = \int_{\Omega}0*f_1(x_1,x_2,x_3)dx+\int_{\Omega}0*f_2(x_1,x_2,x_3)dy + \\ + \int_{\Omega}(-1)*f_3(x_1,x_2,x_3)dz \\ (w,h)=(w,0)=0 \)
Moreover, we assume Young's modulus \( E=1[GPa] \)
and \( \mu=0.3 \) czyli
\( D = \frac{1}{(1.6)(0.4) } \begin{bmatrix} 0.7 & 0.3 & 0.3 & 0 & 0 & 0 \\ 0.3 & 0.7 & 0.3 & 0 & 0 & 0 \\ 0.3 & 0.3 & 0.7 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.2 & 0 & 0\\ 0 & 0 & 0& 0 & 0.2 & 0 \\ 0 & 0 & 0 & 0 & 0& 0.2 \\ \end{bmatrix} [GPa] \)
Then the surface of the platform and each of the five bases can be treated as a separate group of elements and described by the following node vectors
Platform:
[0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 16] x [0 0 0 1 1 1] x [0 0 0 1 1 1]
Supports:
[0 0 0 1 1 1] x [0 0 0 1 1 1] x [0 0 0 10 10 10]
[15 15 15 16 16 16] x [15 15 15 16 16 16] x [0 0 0 10 10 10]
[0 0 0 1 1 1] x [15 15 15 16 16 16] x [0 0 0 10 10 10]
[15 15 15 16 16 16] x [0 0 0 1 1 1] x [0 0 0 10 10 10]
[7 7 7 8 8 8] x [7 7 7 8 8 8] x [0 0 0 10 10 10]
Each of the components of the vector strain field is approximated by a linear combination of the B-spline function on individual groups of elements
\( u^1(x_1,x_2,x_3) = \sum_{i,j,k=1,...,N_x,N_y,N_z} u^1_{i,j,k} B^{x_1}_{i,p}(x_1)B^{x_2}_{j,p}(x_2)B^{x_3}_{k,p }(x_3) \\ u^2(x_1,x_2,x_3) = \sum_{i,j,k=1,...,N_x,N_y,N_z} u^2_{i,j,k} B^{x_1}_{i,p}(x_1)B^{x_2}_{j,p}(x_2)B^{x_3}_{k,p }(x_3) \\ u^3(x_1,x_2,x_3) = \sum_{i,j,k=1,...,N_x,N_y,N_z} u^3_{i,j,k} B^{x_1}_{i,p}(x_1)B^{x_2}_{j,p}(x_2)B^{x_3}_{k,p }(x_3) \)
In our problem of linear elasticity, we simultaneously solve the three components of the displacement field and they fit together into one global matrix.
We put the discretizations into the weak formulation, and behind the testing functions
\( in \) we also take B-spline functions, generate a system of equations, solve it with an exact solver and obtain the solution shown in Fig. 2. The figure will show the norm from deformations, defined as \( \| u^1(x_1,x_2,x_3)^2+u^2(x_1,x_2,x_3)^2+u^3(x_1,x_2,x_3)^2 \| \). The obtained solution is not symmetrical, because the platform also has, apart from four supports located in the corners, a fifth support located asymmetrically.

Figure 2: Norm of platform deformations.