Elasticity theory, extracting the boundary conditions

[c0der]

Homework Statement

I am trying to extract the constant of integration after integrating the following stress equation (I got this by solving a system of ODEs for the upper and lower tablets (attached), quite tedious to paste it all here):

σ = β * sinh(λx/2) * cosh( λ(L-x)/2 )

Where σ is the stress in the upper tablet (see attached) which is Edu₁/dx
β = σ₀ / sinh(λL/2)

Homework Equations

The Attempt at a Solution

Separating the variables and integrating the above stress equation

u₁ = β/2E * [ 1/λ*cosh(λL/2 - λx) + sinh(λL/2)x + C ]

Now, what displacement boundary condition can I apply here to extract that constant? Nothing is evident from the attached image. For the lower tablet (attached), this is simple as it's attached to a roller, and I managed to extract the constant of integration and a solution to the displacement of the lower tablet.

 

[KataKoniK]

you can try to approach this problem by first understanding the physical significance of the constant of integration. In this case, it represents the arbitrary displacement of the upper tablet at the beginning of the system. This means that the value of the constant will depend on the initial conditions of the system and cannot be determined solely from the given equation.

To find the appropriate boundary condition, you can consider the physical constraints of the system. For example, if the upper tablet is fixed at one end and free at the other, the boundary condition would be that the displacement at the fixed end is zero. If the upper tablet is fixed at both ends, the boundary condition would be that the displacement is equal at both ends.

Once you have determined the appropriate boundary condition, you can substitute it into the equation and solve for the constant of integration. Keep in mind that this may require some algebraic manipulation and/or the use of other equations or physical principles.

In summary, extracting the constant of integration requires a thorough understanding of the physical system and the application of appropriate boundary conditions. With careful analysis and problem-solving skills, you can successfully determine the value of the constant and complete your solution.