516
9 Variational Principles in Mechanics and Their Applications
Equations (9.1.4) are called the equation of strain compatibility or compatibility equation, they are also called the condition of deformation compatibility or
compatibility condition.
9.1.5 Work and Energies
The product of the force acting on object and the displacement produced along the
acting direction of the force is called the work. The work done by the force acting on
the object on an arbitrary real displacement is called the actual work or real work;
The work done on an arbitrary virtual displacement is called the virtual work; The
work done on the infinitesimal real displacement or virtual displacement is called
the elementary work.
According to the theory of elastic mechanics, the stress tensor at a point can be
written as the matrix form
σ = [σ i j ] =
⎡
⎣
σ 11 σ 12 σ 13
σ 21 σ 22 σ 23
σ 31 σ 32 σ 33
⎤
⎦ =
⎡
⎣
σ xx σ xy σ xz
σ yx σ yy σ yz
σ zx σ zy σ zz
⎤
⎦ =
⎡
⎣
σ xx τ xy τ xz
τ yx σ yy τ yz
τ zx τ zy σ zz
⎤
⎦
(9.1.5)
The stress tensor is the symmetric tensor.
Let an elementary hexahedron under the action of the stresses σ x , σ y , σ z , τ yz , τ zx ,
τ xy , accordingly produce the strains ε x , ε y , ε z , γ yz , γ zx , γ xy , if the stresses have the
increments δσ x , δσ y , δσ z , δτ yz , δτ zx , δτ xy , then which will cause the corresponding
strain increment δε x , δε y , δε z , δγ yz , δγ zx , δγ xy . Now calculate the work done by the
stresses.
The work done by the normal force σ x dydz corresponds to the elongation δε x dx
produced by the force, the work done by the force can be calculated σ x δε x dxdydz.
A pair of shear forces τ zx dxdy can consist of a couple, the moment of couple is
τ zx dxdydz, when an object has an arbitrary virtual displacement, the work done
by couple is τ zx δγ zx dxdydz. Similarly the work done by the rest external force
components can be written, thus, the virtual work of the infinitesimal hexahedron on
the virtual displacement is
δW = (σ x δε x + σ y δε y + σ z δε z + τ yz δγ yz + τ zx δγ zx + τ xy δγ xy )dxdydz
= σ i j δε i j dxdydz = δW 0 dxdydz
(9.1.6)
where
δW 0 = σ x δε x + σ y δε y + σ z δε z + τ yz δγ yz + τ zx δγ zx + τ xy δγ xy = σ i j δε i j (9.1.7)
9 Variational Principles in Mechanics and Their Applications
Equations (9.1.4) are called the equation of strain compatibility or compatibility equation, they are also called the condition of deformation compatibility or
compatibility condition.
9.1.5 Work and Energies
The product of the force acting on object and the displacement produced along the
acting direction of the force is called the work. The work done by the force acting on
the object on an arbitrary real displacement is called the actual work or real work;
The work done on an arbitrary virtual displacement is called the virtual work; The
work done on the infinitesimal real displacement or virtual displacement is called
the elementary work.
According to the theory of elastic mechanics, the stress tensor at a point can be
written as the matrix form
σ = [σ i j ] =
⎡
⎣
σ 11 σ 12 σ 13
σ 21 σ 22 σ 23
σ 31 σ 32 σ 33
⎤
⎦ =
⎡
⎣
σ xx σ xy σ xz
σ yx σ yy σ yz
σ zx σ zy σ zz
⎤
⎦ =
⎡
⎣
σ xx τ xy τ xz
τ yx σ yy τ yz
τ zx τ zy σ zz
⎤
⎦
(9.1.5)
The stress tensor is the symmetric tensor.
Let an elementary hexahedron under the action of the stresses σ x , σ y , σ z , τ yz , τ zx ,
τ xy , accordingly produce the strains ε x , ε y , ε z , γ yz , γ zx , γ xy , if the stresses have the
increments δσ x , δσ y , δσ z , δτ yz , δτ zx , δτ xy , then which will cause the corresponding
strain increment δε x , δε y , δε z , δγ yz , δγ zx , δγ xy . Now calculate the work done by the
stresses.
The work done by the normal force σ x dydz corresponds to the elongation δε x dx
produced by the force, the work done by the force can be calculated σ x δε x dxdydz.
A pair of shear forces τ zx dxdy can consist of a couple, the moment of couple is
τ zx dxdydz, when an object has an arbitrary virtual displacement, the work done
by couple is τ zx δγ zx dxdydz. Similarly the work done by the rest external force
components can be written, thus, the virtual work of the infinitesimal hexahedron on
the virtual displacement is
δW = (σ x δε x + σ y δε y + σ z δε z + τ yz δγ yz + τ zx δγ zx + τ xy δγ xy )dxdydz
= σ i j δε i j dxdydz = δW 0 dxdydz
(9.1.6)
where
δW 0 = σ x δε x + σ y δε y + σ z δε z + τ yz δγ yz + τ zx δγ zx + τ xy δγ xy = σ i j δε i j (9.1.7)