8.3 Linear Theory for Growing Beams and Plates
411
To include growth, we linearize the 1D decomposition
λ = λ
∗ G,
in which λ is the total longitudinal stretch ratio, λ ∗ is the elastic stretch ratio, and G
is the growth ratio. For small strain, substituting the relations [see Eq. (3.30) 2 ]
λ = 1 +
λ
∗
= 1 +
∗
G = 1 + g
into the expression for λ, expanding the right-hand side, and neglecting nonlinear
terms give
=
∗
+ g .
(8.7)
In this expression, , ∗ , and g are the linear total strain, elastic strain, and
growth strain, respectively, at an arbitrary point in the beam. In the linear theory,
therefore, the total deformation is obtained by adding strain components, rather than
multiplying deformation gradients, as in the nonlinear theory. Note also that g > 0
for positive growth, and g < 0 for atrophy or contraction. Moreover, the linear
strains must be greater than −1 since stretch and growth ratios must be positive.
Force and Moment Resultants For a thin beam, it can be shown that
|σ y |, |σ xy | << |σ x |. Therefore, since σ z = 0, the state of stress in the beam
is approximately 1D, and the axial stress depends on the elastic strain through
Hooke’s law
σ x ≡ σ = EE
∗
= E(( − g )
= E(( 0 + yκ − g ),
(8.8)
where Eq. (8.6) has been used and E is Young’s modulus.
Consider now a differential area element dA = b dy located a distance y from
the middle surface in an arbitrary cross section (Fig. 8.6b). The force acting on the
element is σ dA, and the total normal force and bending moment acting on the cross
section are
N =
A
σ dA = b
h/2
−h/2
σ dy
M =
A
σy dA = b
h/2
−h/2
σy dy.
(8.9)