112
4 Stress–Strain Relations
dU =
1
2
τ xy γ xy dV
(4.37)
The results are analogous to Eqs. (4.34), and (4.37) can be written for any other
pair of stress and strain components (e.g. σ y and ε y or τ yz and γ yz ) whenever the
stress component involved is the only stress acting on the element.
Finally, we consider a general state of stress in which all six stress components
are present. The corresponding deformation will in general involve all six strain
components. The total strain energy stored in the element when the final stresses are
σ x , σ y , σ z , τ xy , τ yz , τ zx and the final strains are ε x , ε y , ε z , γ xy , γ yz , γ zx is thus
dU =
1
2
(σ x ε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx )dV
(4.38)
In general, the final stresses and strains vary from point to point in the body. The
strain energy stored in the entire body is obtained by integrating Eq. (4.32) over the
volume of the body.
U =
1
2
v
(σ x ε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx )dV
(4.39)
The above formula for strain energy applies to small deformations of any linearly
elastic body.
4.5 Boundary Conditions
The boundary conditions are specified in terms of surface forces on certain boundaries
of a body to solve problems in continuum mechanics. When the stress components
vary over the volume of the body, they must be in equilibrium with the externally
applied forces on the boundary of the body. Thus, the external forces may be regarded
as a continuation of internal stress distribution.
Consider a two-dimensional body as shown in Fig. 4.3
Take a small triangular prism ABC, so that the side BC coincides with the boundary
of the plate. At a point P on the boundary, the outward normal is n. Let X and Y be
the components of the surface forces per unit area at this point of boundary. X and
Y must be a continuation of the stresses σ x , σ y and τ xy at the boundary. Now, using
Cauchy’s equation, we have
T x = X = σ x l + τ xy m
T y = Y = τ xy l + σ y m
(a)
in which l and m are the direction cosines of the normal n to the boundary.
4 Stress–Strain Relations
dU =
1
2
τ xy γ xy dV
(4.37)
The results are analogous to Eqs. (4.34), and (4.37) can be written for any other
pair of stress and strain components (e.g. σ y and ε y or τ yz and γ yz ) whenever the
stress component involved is the only stress acting on the element.
Finally, we consider a general state of stress in which all six stress components
are present. The corresponding deformation will in general involve all six strain
components. The total strain energy stored in the element when the final stresses are
σ x , σ y , σ z , τ xy , τ yz , τ zx and the final strains are ε x , ε y , ε z , γ xy , γ yz , γ zx is thus
dU =
1
2
(σ x ε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx )dV
(4.38)
In general, the final stresses and strains vary from point to point in the body. The
strain energy stored in the entire body is obtained by integrating Eq. (4.32) over the
volume of the body.
U =
1
2
v
(σ x ε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx )dV
(4.39)
The above formula for strain energy applies to small deformations of any linearly
elastic body.
4.5 Boundary Conditions
The boundary conditions are specified in terms of surface forces on certain boundaries
of a body to solve problems in continuum mechanics. When the stress components
vary over the volume of the body, they must be in equilibrium with the externally
applied forces on the boundary of the body. Thus, the external forces may be regarded
as a continuation of internal stress distribution.
Consider a two-dimensional body as shown in Fig. 4.3
Take a small triangular prism ABC, so that the side BC coincides with the boundary
of the plate. At a point P on the boundary, the outward normal is n. Let X and Y be
the components of the surface forces per unit area at this point of boundary. X and
Y must be a continuation of the stresses σ x , σ y and τ xy at the boundary. Now, using
Cauchy’s equation, we have
T x = X = σ x l + τ xy m
T y = Y = τ xy l + σ y m
(a)
in which l and m are the direction cosines of the normal n to the boundary.