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Stress, being an artificial concept, cannot be experimentally measured. There are many experimental techniques, though, which can be used to measure strain. Thus, if the relationship between stress and strain is known, the stress state at a point can be calculated after the state of strain has been measured.
We define the principal strains as the strains in the direction of the principal stresses. It is true the the shear strains are zero, just as the shear stresses are zero, on the faces of an element aligned in the principal directions. From equation [2-16] the three principal strains corresponding to a state of unixial stress are
epsilon_1 = sig_1 / E,
epsilon_2 = - mu * epsilon_1 and
epsilon_3 = - mu * epsilon_1 [2-18]
The minus sign is used to indicate compressive strains.
Note that, while the stress state is uniaxial, the strain state
is triaxial.
For the case of biaxial stress we give sig_1 and sig_2 prescribed values and let sig_3 be zero. The principal strains can be found from equation [2-16] if we imagine each principal stress to be acting separately and then combine the results by superposition. This gives
epsilon_1 = sig_1 / E - mu * sig_2 / E,
epsilon_2 = sig_2 / E - mu * sig_1 / E and
epsilon_3 = - mu * sig_1 / E - mu * sig_2 / E [2-19]
Equations [2-19] give the principal strains in terms of the
principal stresses. But the usual situation is the reverse.
To solve for sig_1, multiply epsilon_2 by mu, and add the
first two equations together. This produces
epsilon_1 + mu * epsilon_2 =
sig_1/E - mu*sig_2/E + mu*sig_2/E - mu^2*sig_1/E
Solving for sig_1 gives
sig_1 = E*(epsilon_1+mu*epsilon_2)/(1-mu^2) [2-20]
Similarly,
sig_2 = E*(epsilon_2+mu*epsilon_1)/(1-mu^2) [2-21]
In solving these equations, remember that a tensile stress or
strain is treated as positive and that compression is negative.
The state of stress is said to be triaxial when none of the three principal stresses is zero. For this, the principal strains are
epsilon_1 = sig_1/E - mu*sig_2/E - mu*sig_3/E,
epsilon_2 = sig_2/E - mu*sig_1/E - mu*sig_3/E and
epsilon_3 = sig_3/E - mu*sig_1/E - mu*sig_2/E [2-22]
In terms of the strains, the principal stresses are
sig_1 = (E*epsilon_1*(1-mu)+mu*E*(epsilon_2+epsilon_3))/
(1-mu-2*mu^2) ,
sig_2 = (E*epsilon_2*(1-mu)+mu*E*(epsilon_1+epsilon_3))/
(1-mu-2*mu^2) and
sig_3 = (E*epsilon_3*(1-mu)+mu*E*(epsilon_1+epsilon_2))/
(1-mu-2*mu^2)
[2-23]
Values of Poisson's ratio for various materials are given
in appendix Table A-07.