12

Computational mechanics

The algebraic diﬃculties of analytic mechanical solutions are rather over-

whelming and become insurmountable when solving real-world problems. Com-

putational mechanics is based on the discretization of the geometric continuum

and describing its physical behavior in terms of generalized coordinates. Its

focus is on computing numerical solutions to practical problems of engineering

mechanics.

12.1 Three-dimensional elasticity

One of the fundamental concepts necessary to understanding continuum me-

chanical systems is a generic treatment of elasticity described in detail in the

classical reference of the subject [17]. When an elastic continuum undergoes

a one-dimensional deformation, like in the case of the bar discussed in Section

11.3, Young’s modulus was adequate to describe the changes.

For a general three-dimensional elastic continuum we need another coeﬃ-

cient, introduced by Poisson, to capture the three-dimensional elastic behav-

ior. Poisson’s ratio measures the contraction of the cross-section while an

object such as a beam is stretched. The ratio ν is deﬁned as the ratio of the

relative contraction and the relative elongation:

ν = −

dr

r

/

dl

l

.

Here a beam with circular cross-section and radius r is assumed. Poisson’s

ratio is in the range of zero to 1/2 and expresses the compressibility of the

material. The two constants are also often related as

μ =

E

2(1 + ν)

,

and

λ =

Eν

(1 + ν)(1 − 2ν)

.

177

178 Applied calculus of variations for engineers

Here μ and λ are the so-called Lam´e constants.

In a three-dimensional elastic body, the elasticity relations could vary sig-

niﬁcantly. Let us consider isotropic materials, whose elastic behavior is inde-

pendent of the material orientation. In this case Young’s modulus is replaced

by an elasticity matrix whose terms are only dependent on the Lam´e constants

as follows

D =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

λ +2μλ λ000

λλ+2μλ000

λλλ+2μ 000

000μ 00

0000μ 0

00000μ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

Viewing an inﬁnitesimal cube of the three-dimensional body, there are six

stress components on the element,

σ

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

σ

x

σ

y

σ

z

τ

yz

τ

xz

τ

xy

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

The ﬁrst three are normal and the second three are shear stresses. There are

also six strain components

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

x

y

z

γ

yz

γ

xz

γ

xy

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

The ﬁrst three are extensional strains and the last three are rotational strains.

The stress-strain relationship is described by the generalized Hooke’s law as

σ

= D.

This will be the fundamental component of the computational techniques for

elastic bodies. Let us further designate the location of an interior point of the

elastic body with

r

(x, y, z)=xi + yj + zk =

⎡

⎣

x

y

z

⎤

⎦

,

Computational mechanics 179

and the displacements of the point with

u

(x, y, z)=ui + vj + wk =

⎡

⎣

u

v

w

⎤

⎦

.

Then the following strain relations hold:

x

=

∂u

∂x

,

y

=

∂v

∂y

,

and

z

=

∂w

∂z

.

These extensional strains manifest the change of rate of the displacement of

an interior point of the elastic continuum with respect to the coordinate di-

rections.

The rotational strains are computed as

γ

yz

=

∂v

∂z

+

∂w

∂y

,

γ

xz

=

∂u

∂z

+

∂w

∂x

,

and

γ

xy

=

∂u

∂y

+

∂v

∂x

.

These terms deﬁne the rate of change of the angle between two lines crossing

at the interior point that were perpendicular in the un-deformed body and

get distorted during the elastic deformation.

The strain energy contained in the three-dimensional elastic continuum is

E

s

=

1

2

V

σ

T

dV =

1

2

V

σ

x

σ

y

σ

z

τ

yz

τ

xz

τ

xy

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

x

y

z

γ

yz

γ

xz

γ

xy

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

dV.

We will also consider distributed forces acting at every point of the volume

(like the weight of the beam in Section 11.3), described by

180 Applied calculus of variations for engineers

f

= f

x

i + f

y

j + f

z

k =

⎡

⎣

f

x

f

y

f

z

⎤

⎦

.

The work of these forces is based on the displacements they caused at the

certain points and computed as

W =

V

u

T

fdV. =

V

uvw

⎡

⎣

f

x

f

y

f

z

⎤

⎦

dV.

The above two energy components constitute the total potential energy of the

volume as

E

p

= E

s

− W.

In order to evaluate the dynamic behavior of the three-dimensional body,

the kinetic energy also needs to be computed. Let the velocities at every point

of the volume be described by

˙u

(x, y, z)= ˙ui +˙vj +˙wk =

⎡

⎣

˙u

˙v

˙w

⎤

⎦

.

With a mass density of ρ, assumed to be constant throughout the volume, the

kinetic energy of the body is

E

k

=

1

2

ρ

V

˙u

T

˙udV.

We are now in the position to write the variational statement describing

the equilibrium of the three-dimensional elastic body:

I(u

(x, y, z)) =

V

(E

k

− E

p

)dV = extremum,

which is of course Hamilton’s principle.

The unknown displacement solution of the body at every (x, y, z)pointis

the subject of the computational solution discussed in the next sections.

12.2 Lagrangian formulation

The equations will be obtained by ﬁnding an approximate solution of the vari-

ational problem based on the total energy of the system as follows. For

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