A cube of the isotropic, linear elastic epoxy resin E=2GPa, ν=0.3, α=50×10−6K−1 at 20∘C is unloaded. It is then heated to 60∘C.
What is the corresponding strain matrix?
The temperature is held constant at 60∘C. What stress is required to reduce all the components of the strain matrix to zero?
σ (in MPa):
The temperature is held constant at 60∘ C. What stress is required to reduce all the components of the strain matrix to zero?
σ (in MPa ): -10 MPa
åx=åy=åz=o.oo2
ãx=ãy=ãz=0
I believe that your solution (anonymous) is correct so
ó=(epsilon*E)/(1-2*nu)= 10MPa
The stress is compressive so -10MPa
Do you agree?
e=a*ÄT=0.002
what about ó??
epsilon_x=sigma_x/E-nu*sigma_y/E-nu*sigma_z/E
sigma_x=sigma_y=sigma_z=sigma
epsilon_x=epsilon_y=epsilon_z=epsilon
so...
epsilon=sigma/E-nu*sigma/E-nu*sigma/E
epsilon=(sigma/E)*(1-2*nu)
=> sigma = (epsilon*E)/(1-2*nu)
You want the cube to shrink back to its original position, so plus -0.002 in to the formula. The stress should be negative (compressive).
Is this solution correct?
To find the corresponding strain matrix, we first need to calculate the change in temperature. The change in temperature is given by ΔT = T2 - T1, where T1 is the initial temperature and T2 is the final temperature. In this case, T1 = 20°C and T2 = 60°C.
ΔT = 60°C - 20°C = 40°C
Next, we need to find the coefficient of thermal expansion (α) for the epoxy resin. Given that α = 50 × 10^(-6) K^(-1), we can use this value in the formula:
ΔL = L0 × α × ΔT
where ΔL is the change in length, L0 is the initial length, and ΔT is the change in temperature.
For a cube, all sides are equal, so we can consider the change in length (ΔL) for one side.
ΔL = L0 × α × ΔT
= L0 × (50 × 10^(-6) K^(-1)) × 40°C
Now, we can calculate the change in length for one side. Let's assume the initial length of one side is L0 = 1 meter.
ΔL = (1 meter) × (50 × 10^(-6) K^(-1)) × (40°C)
= 0.000002 meters
Since strain is defined as the change in length divided by the initial length, we can calculate the strain as:
Strain = ΔL / L0
= 0.000002 meters / 1 meter
= 0.000002
Since this is a cubic material, the strain occurs in all three directions (x, y, and z). Therefore, the corresponding strain matrix would be:
Strain Matrix:
[0.000002 0 0]
[0 0.000002 0]
[0 0 0.000002]
To find the stress required to reduce all components of the strain matrix to zero, we need to consider Hooke's Law for isotropic, linear elastic materials:
σ = E * ε
where σ is the stress, E is the elastic modulus, and ε is the strain.
Since all components of the strain matrix are zero, the required stress (σ) to achieve this would also be zero.
εx=
εy=
εz=
γyz=
γxz=
γxy