14
1 Summary of Elasticity Theory: Basic Concepts
Fig. 1.7 Homogeneous
stressed state
whereas the index k coincides with the designation of the coordinate axis parallel to
the respective normal components of the stress vector. In the case of a homogeneous
stressed state, normal stresses on opposite faces are equal in magnitude.
Two other components of the stress vector will be located in the face plane. They
are called tangential stresses. Due to the law of twoness, tangential stresses are fully
defined by three values: τ xy , τ yz , τ zx . The first of the indexes in the designation
of tangential stresses indicates the axis in parallel to which this component of the
stress vector acts, while the second index shows the normal line to the area under
consideration.
Figure 1.7 shows the components of the stress vector on three (visible) areas only.
On each of three other (non-visible) areas, the directions of stress vector components
will be opposite to those that have place on the opposite (visible) area.
Let us describe the rules of signs for stresses. Let us assume that the direction
of the external normal line to the surface of the element under consideration of a
body coincides with the direction of the coordinate axis. In this case, stress vector
components are deemed positive when their direction coincides with the coordinate
direction. Alternatively, when the external normal line to the surface is directed
opposite to the coordinate axis direction, stress vector components are directed
opposite to the directions of coordinate axes.
1.13 Generalized Hooke’s Law
Homogeneous deformation is characterized by three relative elongations (ε x , ε y , ε z )
in the direction of coordinate axes and three shear deformations of an element
parallel to the coordinate axes (γ xy , γ yz , γ zx ). In an elastic body, these deformations
are associated with stresses by the generalized Hooke’s law:
1 Summary of Elasticity Theory: Basic Concepts
Fig. 1.7 Homogeneous
stressed state
whereas the index k coincides with the designation of the coordinate axis parallel to
the respective normal components of the stress vector. In the case of a homogeneous
stressed state, normal stresses on opposite faces are equal in magnitude.
Two other components of the stress vector will be located in the face plane. They
are called tangential stresses. Due to the law of twoness, tangential stresses are fully
defined by three values: τ xy , τ yz , τ zx . The first of the indexes in the designation
of tangential stresses indicates the axis in parallel to which this component of the
stress vector acts, while the second index shows the normal line to the area under
consideration.
Figure 1.7 shows the components of the stress vector on three (visible) areas only.
On each of three other (non-visible) areas, the directions of stress vector components
will be opposite to those that have place on the opposite (visible) area.
Let us describe the rules of signs for stresses. Let us assume that the direction
of the external normal line to the surface of the element under consideration of a
body coincides with the direction of the coordinate axis. In this case, stress vector
components are deemed positive when their direction coincides with the coordinate
direction. Alternatively, when the external normal line to the surface is directed
opposite to the coordinate axis direction, stress vector components are directed
opposite to the directions of coordinate axes.
1.13 Generalized Hooke’s Law
Homogeneous deformation is characterized by three relative elongations (ε x , ε y , ε z )
in the direction of coordinate axes and three shear deformations of an element
parallel to the coordinate axes (γ xy , γ yz , γ zx ). In an elastic body, these deformations
are associated with stresses by the generalized Hooke’s law: