12
1 Summary of Elasticity Theory: Basic Concepts
tg OA
B
= tg
90 ◦ − γ
2
=
1 − tg
γ
2
1 + tg
γ
2
.
Comparing the last result with formula (1.6), we get
tg
γ
2
=
1 + ν
E
p.
(1.7)
Let us find now tangential stresses acting on the faces of square ABCD. To do
it, let us compare the equilibrium condition of the element BCL shown in Fig. 1.5a.
We notice that
BCL = CBL = 45
◦ , BL = LC =
BC
√
2
(1.8)
and zero the sum of projections of all forces to the direction BC, we get τ = p, and
formula (1.7), provided low deformation
tg
γ
2
≈
γ
2
,
can be written as
γ =
τ
G
(1.9)
that designates
G =
E
2(1 + ν)
.
(1.10)
The formula (1.9) expresses the Hooke’s law in shear. It defines increment (γ ) of
the straight angle as a result of action of tangential stresses τ as shown in Fig. 1.5b.
This increment is called shear deformation, and the coefficient G included in
formula (1.9) is called the shear modulus.
1.11 Law of Twoness of Tangential Stresses
Assume that a body shaped as a rectangular parallelepiped (Fig. 1.6) is exposed to
the action of only tangential forces in the conditions of a homogeneous stressed state
[7].
Provided that the projections of forces onto coordinate axes are equal to zero xy,
it goes that τ
x = τ x , τ
y = τ y , e. g., tangential stresses on parallel faces are equal
1 Summary of Elasticity Theory: Basic Concepts
tg OA
B
= tg
90 ◦ − γ
2
=
1 − tg
γ
2
1 + tg
γ
2
.
Comparing the last result with formula (1.6), we get
tg
γ
2
=
1 + ν
E
p.
(1.7)
Let us find now tangential stresses acting on the faces of square ABCD. To do
it, let us compare the equilibrium condition of the element BCL shown in Fig. 1.5a.
We notice that
BCL = CBL = 45
◦ , BL = LC =
BC
√
2
(1.8)
and zero the sum of projections of all forces to the direction BC, we get τ = p, and
formula (1.7), provided low deformation
tg
γ
2
≈
γ
2
,
can be written as
γ =
τ
G
(1.9)
that designates
G =
E
2(1 + ν)
.
(1.10)
The formula (1.9) expresses the Hooke’s law in shear. It defines increment (γ ) of
the straight angle as a result of action of tangential stresses τ as shown in Fig. 1.5b.
This increment is called shear deformation, and the coefficient G included in
formula (1.9) is called the shear modulus.
1.11 Law of Twoness of Tangential Stresses
Assume that a body shaped as a rectangular parallelepiped (Fig. 1.6) is exposed to
the action of only tangential forces in the conditions of a homogeneous stressed state
[7].
Provided that the projections of forces onto coordinate axes are equal to zero xy,
it goes that τ
x = τ x , τ
y = τ y , e. g., tangential stresses on parallel faces are equal