Energy methods in elasticity

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Energy Methods/Variational Principles[edit | edit source]


  • 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 | edit source]

Functional[edit | edit source]

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 | edit source]

Find such that

is a minimum.

Variation[edit | edit source]


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

Necessary Condition : Euler Equation[edit | edit source]

A necessary condition that minimizes is that the Euler equation

is satisfied and



and is small and is arbitrary.

Imposed BCs[edit | edit source]

The imposed BCs are the conditions

These are automatically satisfied.

Natural BCs[edit | edit source]

The natural BCs are the conditions

Stationary Functions[edit | edit source]

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

Taking Variations[edit | edit source]

Suppose that is a functional with

Suppose that is a small variation of that satisfies

Then the variation of is


The variation of is


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

Lagrange Multipliers[edit | edit source]

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

More on Strain Energy Density[edit | edit source]

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 | edit source]

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


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

Example[edit | edit source]

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 | edit source]

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


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 | edit source]

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

The Principle of Stationary Potential Energy[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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


In index notation,

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

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 | edit source]


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

Minimum[edit | edit source]


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

Observations on Uniqueness and Existence of Solutions[edit | edit source]

  • 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 | edit source]

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


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


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%.

Related Content[edit | edit source]

Introduction to Elasticity