References
153
Fig. 3.28 Elastic bar with
attached mass (Problem 3.21)
m
f(t)
x(t)
where c = 20 Pa. The deformation gradient tensor at a point in a body is
F = 1.5 e 1 e 1 + 2 e 2 e 2 + 0.5 e 1 e 2 + e 3 e 3 .
(a) Compute the first Piola-Kirchhoff stress vector on a plane that passes
through this point and is normal to the vector B = e 1 − e 2 + 2e 3 in the
undeformed body.
(b) During deformation, the vector B is transformed into the vector b.
Compute the Cauchy stress vector on a plane that passes through the same
point and is normal to b.
3.20 Using equations developed in this chapter and including only mechanical
energy, show that the energy balance relation (3.163) can be derived by dotting
the equation of motion (3.145) with the velocity v and integrating over the
deformed volume V .
3.21 An elastic bar with Young’s modulus E, length L, and cross-sectional area
A is fixed at one end and attached to a mass m at the other end. A force
f (t) acts on the mass, which slides on a frictionless surface (Fig. 3.28). For
small deformation, the strain energy in the bar per unit volume is given by
W =
1
2 EE 2 in terms of the linear axial strain . Neglecting the mass of the
bar, show that the energy balance relation (3.163) for the entire system yields
the equation of motion in the form
m ¨
x + kx = f (t),
where x is the displacement of the mass, and k represents the equivalent extensional stiffness of the bar. Determine k and compute the natural frequency
ω =
√
k/m for the unloaded bar.
References
Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures.
Wiley, Chichester
Blatz PD and Ko WL (1962) Application of finite elasticity to the deformation of rubbery materials.
Trans. Soc. Rheology 6:223–251
Chadwick P (1976). Continuum mechanics: concise theory and problems. Wiley, New York
Chandrasekharaiah D, Debnath L (2014) Continuum mechanics. Elsevier, Amsterdam
Eringen AC (1962) Nonlinear theory of continuous media. McGraw-Hill, New York
Eringen AC (1980) Mechanics of continua. R. E. Krieger Pub. Co, Huntington
153
Fig. 3.28 Elastic bar with
attached mass (Problem 3.21)
m
f(t)
x(t)
where c = 20 Pa. The deformation gradient tensor at a point in a body is
F = 1.5 e 1 e 1 + 2 e 2 e 2 + 0.5 e 1 e 2 + e 3 e 3 .
(a) Compute the first Piola-Kirchhoff stress vector on a plane that passes
through this point and is normal to the vector B = e 1 − e 2 + 2e 3 in the
undeformed body.
(b) During deformation, the vector B is transformed into the vector b.
Compute the Cauchy stress vector on a plane that passes through the same
point and is normal to b.
3.20 Using equations developed in this chapter and including only mechanical
energy, show that the energy balance relation (3.163) can be derived by dotting
the equation of motion (3.145) with the velocity v and integrating over the
deformed volume V .
3.21 An elastic bar with Young’s modulus E, length L, and cross-sectional area
A is fixed at one end and attached to a mass m at the other end. A force
f (t) acts on the mass, which slides on a frictionless surface (Fig. 3.28). For
small deformation, the strain energy in the bar per unit volume is given by
W =
1
2 EE 2 in terms of the linear axial strain . Neglecting the mass of the
bar, show that the energy balance relation (3.163) for the entire system yields
the equation of motion in the form
m ¨
x + kx = f (t),
where x is the displacement of the mass, and k represents the equivalent extensional stiffness of the bar. Determine k and compute the natural frequency
ω =
√
k/m for the unloaded bar.
References
Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures.
Wiley, Chichester
Blatz PD and Ko WL (1962) Application of finite elasticity to the deformation of rubbery materials.
Trans. Soc. Rheology 6:223–251
Chadwick P (1976). Continuum mechanics: concise theory and problems. Wiley, New York
Chandrasekharaiah D, Debnath L (2014) Continuum mechanics. Elsevier, Amsterdam
Eringen AC (1962) Nonlinear theory of continuous media. McGraw-Hill, New York
Eringen AC (1980) Mechanics of continua. R. E. Krieger Pub. Co, Huntington