Energy relations and variational principles

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Energy relations and variational principles

While deriving the constitutive equation system of finite-element method the variations in the principles of mechanics are used, which are based on the energy relations.

Click to show/hide hidden textBasic energy equation

Equation of conservation of mechanical energy (power) of deformed body in the absence of inertia loads states that the power of specific external forces p applied to the outer surface F of deformed body is equal to power developed by stresses σ over the strain rate in the bulk of the deformed body V.

Here i,j = x,y,z, vi - the projections of velocities of surface material points on the coordinate axes. In the coordinate form the expression above has a form:

Any external surface can be represented as a set of two parts:

•surface Fp on which the natural boundary conditions are given in p0

•surface Fv on which the essential boundary conditions are given v0 (with kinematic constraints)

The difference between external and internal forces powers at the surface Fp is called the total power
Click to show/hide hidden text Variational principles

Variational principles in mechanics determine the properties of the system's actual motion under the action of specified forces that allows it to distinguish this motion from kinematically-admissible motion (that satisfies the boundary conditions and constraints). Usually variational principles are formulated in the form of the condition for minimum of some quantity (functional) that depends on one or several functions related to physics of the task.

In variational principles the following terms are used:

•true (actual) displacement, which satisfies boundary conditions, under which the body is in equilibrium

•admissible (kinematically-admissible) displacement, which satisfies boundary condition, but under which the body is not necessary in equilibrium. There exists an infinite number of these kinematically-admissible displacements.

•virtual displacement that represents the difference between kinematically-admissible displacement and true displacement. Displacement variations δu are equal to zero wherein the external restrictions (constraints) for system points motion are given.

As it follows from Lagrange principle, for a body in equilibrium the basic energy equation is true for any system of kinematically-admissible displacements (velocities). Thus, this equation is also true for virtual velocities dV (change of virtual displacements per unit time)

In the left part is written only the external forces work at the surface part Fp, where the kinematic constraints are not specified, because velocity variations are equal to zero for essential boundary conditions.

Here

- strain rate increment for virtual velocities

As far as the specific forces and stresses do not vary:

the variation of strain full power (virtual power) of the deformed body is equal to zero

Total power minimum principle follows from the principle of virtual work (power) and is formulated as follows [Washizu K.(1982)]:

Among all the admissible displacements true displacements supply the total power functional with the stationary value.

It can be said that this value is a minimum one.

Thus, the body shape corresponding to its equilibrium differs from all other forms with its minimum value of total power.

For visco-plastic bodies the Markov's variational principle is true. According to it the functional

takes the minimum value for actual velocity field.

First integral of functional reflects the power of plastic strain, and the second reflects the power of external forces.

Markov's functional is derived directly from the expression for total power of the deformed body when using the Levy–Mises equations as the constitutive equations.

In the matrix form that is used in the FE method, Markov's functionality can be written as follows:

Here {v} and {p} - are the matrix vectors of material points velocities and known specific forces on the external surface, respectively.