Aluminum is a cubic crystal, with
S11=1.59×10−11 m2/N
S12=−0.58×10−11 m2/N
S44=3.52×10−11 m2/N
What are the Young's modulus, Poisson's ratio and shear modulus, corresponding to these values?
Young's modulus E (in GPa):
Poisson's ratio ν:
Shear modulus G (in GPa):
For a single crystal of aluminum, loaded uniaxially along the crystal axis with a stress of σ1=10MPa, what are all of the components of strain?
What is the total elastic strain energy, per unit volume, for the uniaxial stress of 10MPa?
Strain energy (in J/m3):
E = 62.89
p.r = 0.365
G = 24.81
ex = 0.159*10^-3
ey = -0.058*10^-3
ez = -0.058*10^-3
gammaxy,yz,zx = 0
energy = 795
G=23.03
Shear modulus G (in GPa):???
G=28.41
To find the values of Young's modulus (E), Poisson's ratio (ν), and shear modulus (G) using the given values of the compliance (S) constants, we can use the following relationships:
Young's modulus (E) = S11
Poisson's ratio (ν) = -S12/S11
Shear modulus (G) = S44
Given:
S11 = 1.59×10−11 m2/N
S12 = -0.58×10−11 m2/N
S44 = 3.52×10−11 m2/N
Let's calculate each value one by one:
Young's modulus (E):
E = S11 = 1.59×10−11 m2/N
Poisson's ratio (ν):
ν = -S12/S11 = (-0.58×10−11 m2/N)/(1.59×10−11 m2/N)
Shear modulus (G):
G = S44 = 3.52×10−11 m2/N
Now, let's calculate the components of strain for a single crystal of aluminum under uniaxial stress:
Given:
Stress (σ1) = 10 MPa = 10×10^6 Pa
Components of strain (εij) can be calculated using Hooke's law, as follows:
ε11 = σ1/E
ε22 = -(νε11)
ε33 = -(νε11)
ε12 = ε21 = ε23 = ε32 = 0
Total elastic strain energy (U) per unit volume can be calculated using the equation:
U = (1/2)σ1(ε11 + ε22 + ε33)
Let's calculate the strain components and strain energy:
ε11 = (10×10^6 Pa)/(1.59×10−11 m2/N)
ε22 = -(νε11)
ε33 = -(νε11)
ε12 = ε21 = ε23 = ε32 = 0
Now, plug the values of ε11, ε22, ε33 into the equation of strain energy:
U = (1/2)(10×10^6 Pa)[(ε11) + (ε22) + (ε33)]
Finally, substitute the values of the strain components into the equation and calculate U to find the strain energy per unit volume.
I hope this helps!