# Energy methods in elasticity

## Contents

- 1 Energy Methods/Variational Principles
- 2 Some definitions from Variational Calculus
- 3 Taking Variations
- 4 Lagrange Multipliers
- 5 More on Strain Energy Density
- 6 The Principle of Virtual Work
- 7 Energy Minimization Principles
- 7.1 The Principle of Stationary Potential Energy
- 7.2 The Principle of Minimum Potential Energy
- 7.3 Kinematically Admissible Displacement Fields
- 7.4 Potential Energy Functional
- 7.5 Stationary Points and Minimum of the Potential Energy Functional
- 7.6 Stationary Point
- 7.7 Minimum
- 7.8 Observations on Uniqueness and Existence of Solutions

- 8 Example: Approximate Solutions of Torsion Problems

## Energy Methods/Variational Principles[edit]

Examples:

- Principle of Virtual Work.
- Principle of Minimum Potential Energy.
- Principle of Minimum Complementary Energy.
- Hu-Washizu Variational Principle.
- Hellinger-Reissner Variational Principle.

Why ?

- Powerful way of approaching problems in linear elasticity.
- Can be used to derive the governing equations and boundary conditions for special classes of problems.
- Used as the basis of approximate solutions of elasticity problem, e.g., finite element method.
- Can be used to obtain rigorous bounds on the stiffness of elastic structures/solids.

## Some definitions from Variational Calculus[edit]

### Functional[edit]

A functional is basically a function of some other functions.
Let be the displacement. Then the local strain energy density
is a *functional*.

### The Minimization Problem[edit]

Find such that

is a minimum.

### Variation[edit]

Suppose

and equality holds only when . Then is the
*variation* of .

### Necessary Condition : Euler Equation[edit]

A necessary condition that minimizes is that the Euler equation

is satisfied and

and,

where

and is small and is arbitrary.

### Imposed BCs[edit]

The imposed BCs are the conditions

These are automatically satisfied.

### Natural BCs[edit]

The natural BCs are the conditions

### Stationary Functions[edit]

Any that satisfies the necessary conditions make the functional
*stationary* and is said to be a *stationary function*
of the functional.

## Taking Variations[edit]

Suppose that is a functional with

Suppose that is a small variation of that satisfies

Then the *variation* of is

or,

The variation of is

or,

Assuming that is a necessary condition to minimize , we get the same necessary conditions as before.

## Lagrange Multipliers[edit]

If there are additional constraints on minimization, we usually use
*Lagrange Multipliers*.

Suppose the additional constraint is that .
Then, we define a function,

where is the Lagrange multiplier.

Then,

Then, the values that minimize the function subject to the given constraint are given by the equations

## More on Strain Energy Density[edit]

Recall that the strain energy density is defined as

If the strain energy density is path independent, then it acts as a potential for stress, i.e.,

For *adiabatic processes*, is equal to the change in internal energy per unit volume.

For *isothermal processes*, is equal to the Helmholtz free energy per unit volume.

The *natural state* of a body is defined as the state in which the body is in stable thermal equilibrium with no external loads and zero stress and strain.

When we apply energy methods in linear elasticity, we implicitly assume that a body returns to its natural state after loads are removed. This implies that the *Gibb's condition* is satisfied :

## The Principle of Virtual Work[edit]

This principle is used in the derivation of several minimization principles and states that:

If is a state of stress satisfying equilibrium

and the traction boundary condition

Also, if is a displacement field on such that the strain field is given by

then

The converse also holds - and is usually more interesting because it gives us a different way of thinking about equilibrium.

### Example[edit]

If there are jump discontinuities in a body, then what does equilibrium imply ?

Suppose that has a jump discontinuity across a body
along the surface with normal because the materials
on the two sides are different.

We define the equilibrium state to be one that satisfies the principle
of virtual work for all displacement fields.

Now, if the spin tensor is zero, then

If we use the above, and apply the divergence theorem to the virtual work equation we get

For the stress jump to satisfy this equation, we must have

Hence, equilibrium is satisfied when

which means that even though a jump can exist in the stresses, the tractions have to be continuous across the discontinuity.

### Energy as a Functional[edit]

The work done by external forces on a body can be represented as a functional

Taking the variation of , we get

In index notation,

Noting that the external forces and body forces are not functions of , the above equation reduces to

The above expression is called the *external virtual work.*

If we apply the principle of virtual work, with

we get

or,

This is the expression for the *internal virtual work.*

Thus, another form of the principle of virtual work is

Doing the reverse operation, it can be shown that

which relates the strain energy density () to the functional that represents the work done by external forces.

## Energy Minimization Principles[edit]

Developed and explored by Green (1839), Haughton(1849), Kirchhoff (1850), Love (1906), Trefftz (1928) and others.

### The Principle of Stationary Potential Energy[edit]

This principle states that:

- Among all possible
*kinematically admissible displacement fields* - the
*potential energy functional*is rendered*stationary* - by only those that are
*actual*displacement fields.

### The Principle of Minimum Potential Energy[edit]

This principle states that

- If the prescribed traction and body force fields are
*independent*of the deformation - then the
*actual displacement field*makes the*potential energy functional*an absolute*minimum*.

### Kinematically Admissible Displacement Fields[edit]

Consider a body with a boundary with an applied body force field .

Suppose that
displacement BCs are prescribed on the part of
the boundary .

Suppose also that traction BCs
are applied on the portion of the
boundary .

A displacement field is *kinematically admissible* if

- satisfies the displacement boundary conditions on .
- is continuously differentiable, i.e., and .

### Potential Energy Functional[edit]

The *potential energy functional* associated with the
kinematically admissible displacement field is defined as

or,

In index notation,

### Stationary Points and Minimum of the Potential Energy Functional[edit]

What do we mean when we say that we "render the potential energy functional stationary" or "minimum"? Note that the potential energy is a functional of a vector field.

Suppose that the actual displacement field (one that satisfies equilibrium,
compatibility and the boundary conditions) is .

Let be a kinematically admissible variation of , i.e.,

where is a constant.

Then must be a displacement field that is continuously
differentiable and satisfies the boundary conditions

The potential energy functional for is

In index notation,

Expanding and rearranging,

Using the symmetry of the stiffness tensor, we can simplify the above expression and write it it terms of variations of . Thus,

You can check that the first and second variations of turn out to be equal to the expanded terms in equation (1).

### Stationary Point[edit]

If

for all admissible variations , then is a { stationary point} of the functional .

### Minimum[edit]

If

for all admissible variations , then is makes the
functional a *minimum*.

### Observations on Uniqueness and Existence of Solutions[edit]

- The potential energy functional is a global minimum if and only if the displacement field satisfies traction BCs, equilibrium and the displacement BCs.
- Thus, if the potential energy functional actually has a global mininum, then a solution exists and must be unique.
- Displacement boundary value problems do not face the problem of rigid body motions. Therefore, a global minimum always exists for such problems and is unique.
- Traction boundary value problems may not have unique solutions, nor might solutions always exist unless the external loads are in static equilibrium.

## Example: Approximate Solutions of Torsion Problems[edit]

Suppose that we have a cylindrical body of length and an arbitrary cross-section that is subject to equal and opposite torques at the two ends. The displacement field is given by

The traction-free boundary conditions on the lateral surfaces can be given as

The torque BC at the ends can be replaced with displacement BCs

and

Thus, we change the problem from a purely traction boundary value problem to one in which the twist per unit length () is prescribed instead of the applied torque ().

The modified problem is one with zero body force and zero tractions.
Therefore, the potential energy functional reduces to

The stresses and strains for the torsion problem are given by

Therefore, the internal energy is

The potential energy per unit length () is

According to the principle of minimum potential energy, the actual warping function is the one that makes an absolute minimum.

Suppose we are given a warping function of the form

Then, the potential energy per unit length is

If and are the principal axes of inertia, then we have

Hence,

The stationary points of the potential energy functional are given by

Thus, we have,

Thus the best approximation to the warping function is

The above technique is called the *Rayleigh-Ritz method.*

An important observation that should be made at this stage is about the *approximate* nature of the solution.

For cross-sections in which , (e.g., circular or square sections) the best approximation for is . This gives us the exact result for circular cross-sections.

However, for square cross-sections we have an error of nearly 20%.