Chapter 8
Linear Elastic Systems
8.1 General Comments
Let some system of material points be in equilibrium under the action of a defined
load. Due to equilibrium, the resultant of all forces applied to each point of the
system equals zero. Let us give infinitely short shifts to system points and keep
forces between them unchanged. The work of the defined forces applied to any point
of the system will be zero. Hence it follows that for the entire system of material
points, the work of the defined forces in equilibrium equals zero. This means that
the sum of works of internal and external forces equals zero, e.g.
A + W = 0,
(8.1)
where A is the work of external forces and W is the work of internal forces.
In the case of an absolute solid body, displacements occur without deformation,
distances between particles do not change, and internal forces perform no work. In
this case, formula (8.1) shows the principle of possible displacements [3, p. 201].
Let us represent the work of internal forces as a sum of products of parameters
(generalized forces P i ) defining the load and some coefficients depending on
the type and magnitude of displacements. These coefficients are referred to as
generalized displacements (u i ). In a similar way, the work of internal forces at
infinitely short displacements of a loaded system can be represented as follows:
W = − i u i .
(8.2)
The work (8.2) is called [2] virtual. It differs from the actual work of forces P i
performed during loading.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_8
77
Linear Elastic Systems
8.1 General Comments
Let some system of material points be in equilibrium under the action of a defined
load. Due to equilibrium, the resultant of all forces applied to each point of the
system equals zero. Let us give infinitely short shifts to system points and keep
forces between them unchanged. The work of the defined forces applied to any point
of the system will be zero. Hence it follows that for the entire system of material
points, the work of the defined forces in equilibrium equals zero. This means that
the sum of works of internal and external forces equals zero, e.g.
A + W = 0,
(8.1)
where A is the work of external forces and W is the work of internal forces.
In the case of an absolute solid body, displacements occur without deformation,
distances between particles do not change, and internal forces perform no work. In
this case, formula (8.1) shows the principle of possible displacements [3, p. 201].
Let us represent the work of internal forces as a sum of products of parameters
(generalized forces P i ) defining the load and some coefficients depending on
the type and magnitude of displacements. These coefficients are referred to as
generalized displacements (u i ). In a similar way, the work of internal forces at
infinitely short displacements of a loaded system can be represented as follows:
W = − i u i .
(8.2)
The work (8.2) is called [2] virtual. It differs from the actual work of forces P i
performed during loading.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_8
77