1.11 The Weak Form of the Balance of Momentum

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In this section, an alternative form of the balance of momentum will be derived, which will

form the basis for much of the finite element formulation to follow in subsequent chapters.

In the material formulation, the weak form is generally known as the principle of virtual

work and in the spatial description it is known as the virtual power principle. It will be

shown that the strong form deduced so far, Equation (1.288), is completely equivalent to

the weak form by first deriving the weak form from the strong form and subsequently the

strong form from the weak form. General curvilinear coordinates are assumed throughout.

To obtain the equations in rectangular coordinates, replace the covariant differentiation by

partial differentiation.

1.11.1 Formulation of the boundary conditions (material coordinates)

The balance of momentum in material form

PKk

;K

+ ρ0f k = ρ0

D2uk

Dt2 (1.307)

will be supplemented here with boundary conditions. Suppose that the material volume

V0 is surrounded by a surface A0 consisting of internal surfaces A0i , surfaces on which

the displacements are described A0u and surfaces on which the traction is defined, A0t .

Accordingly,

T

+

(N

+

)

+ T

−

(N

−

)

=0 onA0i (1.308)

u = u on A0u (1.309)

T (N) = T (N) on A0t . (1.310)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 39

At internal surfaces the material is connected, but it might change its properties, for

example, due to a change of material. Equation (1.308) is equivalent to Newton’s third

law: action equals reaction. The plus and minus sign denote the two sides of the internal

surface. Since

T (N) = T KNK (1.311)

and N

− = −N

+, Equation (1.308) also reads

(T K

+

− T K

−

)N

+

K

=:         T KN

+

K

= 0. (1.312)

This is also called the traction continuity condition. Note that Equation (1.307) is equivalent

to

PKk

;Kgk

+ ρ0f kgk

= ρ0

D2uk

Dt2 gk (1.313)

and hence,

PKk

;Kg L

k

+ ρ0f kg L

k

= ρ0

D2uk

Dt2 g L

k (1.314)

where g L

k := gk

· GL are called shifters since they move quantities from one coordinate

system into another. Indeed, for a vector v one has

v = vkgk

⇒ v · GL = vkg L

k

⇒ V L = vkg L

k . (1.315)

Accordingly, Equation (1.307) is equivalent to

PKk

;Kg L

k

+ ρ0f L = ρ0

D2uL

Dt2 . (1.316)

1.11.2 Deriving the weak form from the strong form (material

coordinates)

Let us consider an infinitesimal perturbation of the displacement field δu with components

δuk satisfying the geometric boundary conditions in Equation (1.309). Accordingly,

δu =0 onA0u. (1.317)

Taking the scalar product of the vector Equation (1.307) with the one-form δu and integrating

over the material volume leads to

_

V0

_

PKk

;K

+ ρ0

_

f k − D2uk

Dt2

__

δuk dV = 0. (1.318)

Since (the usual differentiation rules also apply to covariant differentiation)

PKk

;Kδuk = (P Kkδuk);K − PKkδuk;K (1.319)

40 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

and applying Cauchy’s theorem, Equation (1.234), one obtains

_

A0

PKkNKδuk dA − _

V0

_

PKkδuk;K − _

ρ0

_

f k − D2uk

Dt2

__

δuk

_ dV = 0 (1.320)

or

_

V0

PKkδuk;K dV = _

A0t

T

K

(N)δUK dA + _

V0

ρ0f KδUK dV − _

V0

ρ0

D2uK

Dt2 δUK dV

(1.321)

since T K+

(N

+

)

+ T K−

(N

−

)

= 0 on A0i , δUK = 0 on A0u and T (N) = T (N) on A0t . Through

the relationship

PKk = SKLxk

,L (1.322)

one obtains

PKkδuk;K = SKLxk

,Lδxk,K = SKLxk

,Lδxm

,Kgkm = SKLδEKL. (1.323)

Indeed,

SKLδEKL = SKLδ _12

xk

,Lxm

,Kgkm − 12

GKL

= 12

SKL(δxk

,Lxm

,K

+ xk

,Lδxm

,K)gkm

= SKLxk

,Lδxm

,Kgkm (1.324)

since both SKL and gkm are symmetric. Notice that covariant differentiation does not apply

to x. Indeed, from

u = o + x − X (1.325)

one obtains

u,K = x,K − GK (1.326)

leading to (see Equation (1.25))

uk

;K

= xk

,K

− gk

K. (1.327)

Concluding, Equation (1.321) can also be written as

_

V0

SKLδEKL dV = _

A0t

T

K

(N)δUK dA + _

V0

ρ0f KδUK dV − _

V0

ρ0

D2UK

Dt2 δUK dV.

(1.328)

The left-hand side is called the internal virtual work, the first term on the right-hand side is the

virtual work due to external tractions, the second term is due to distributed forces and the last

term is due to inertia. Notice that, although all quantities in Equation (1.328) are expressed

in terms of material coordinates, some are defined as a function of their spatial counterparts

such as T (N) and f through T (N)(X, t) dA = t (n)(X, t) da and f K(X, t)GK = f k(X, t)gk.

Hence, both T (N) and f K are a function of the deformation. For example, if a rotating body

expands because of centrifugal loads, f K changes.

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 41

1.11.3 Deriving the strong form from the weak form (material

coordinates)

Starting from the weak form in Equation (1.321) and applying Equation (1.319) and Cauchy’s

theorem one obtains

_

A0

PKkNKδuk dA − _

V0

PKk

;Kδuk dV

− _

A0t

T

k

(N)δuk dA − _

V0

ρ0

_

f k − D2uk

Dt2

_

δuk dV = 0. (1.329)

Since PKkNK = T k

(N), A0 = A0u ∪ A0t ∪ A0i and δu = 0 for A0u, one obtains

_

A0t

_T k

(N)

− T

k

(N)

  δuk dA + _

A0i

_T k+

(N

+

)

+ T k−

(N

−

)

  δuk dA

− _

V0

_

PKk

;K

+ ρ0f k − ρ0

D2uk

Dt2

_

δuk dV = 0. (1.330)

So far we only specified δu to be a virtual displacement field satisfying the geometric

boundary conditions. Now, we require Equation (1.330) to be valid not only for one

special δu but also for any δu satisfying δu = 0 on A0u. Because of the arbitrariness of

δu, the functional analysis density theorem applies (for a proof, the reader is referred to

(Belytschko et al. 2000)) requiring the coefficients of δuk in each term in Equation (1.330)

to be zero. Accordingly,

T k

(N)

− T

k

(N)

=0 onA0t (1.331)

T k+

(N

+

)

+ T k−

(N

−

)

on A0i (1.332)

PKk

;K

+ ρ0f k = ρ0

D2uk

Dt2 on V0 (1.333)

which is the strong form.

1.11.4 The weak form in spatial coordinates

Here again, the starting point is the strong form, Equation (1.281)

(σ klgl),k + ρf lgl

= ρ

Dvl

Dt

gl (1.334)

subject to

t k+

(n+

)

+ t k−

(n−

) on Ai (1.335)

u = u on Au (1.336)

t k

(n)

− t k

(n)

=0 onAt . (1.337)

42 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

In spatial coordinates, a virtual velocity field δv is selected satisfying δv = 0 on Au.

The reason for this will become clear in the derivation. Scalar multiplication yields

_

V

            (σ klgl),k · δv + ρf lgl

· δv dv = _

V

ρ

Dvl

Dt

gl

· δv dv. (1.338)

Since

(σ klgl),k · δv = (σ klgl

· δv),k − σ klgl

· δv,k (1.339)

and

δv,k = (δvmgm),k = δvm;kgm (1.340)

one obtains

_

V

_

(σ klδvl),k − σ klδvl;k + ρ

_

f l − Dvl

Dt

_

δvl

_ dv = 0 (1.341)

or

_

A

σ klnkδvl da − _

V

_

σ klδdkl − ρ

_

f l − Dvl

Dt

_

δvl

_ dv = 0. (1.342)

Indeed, dkl = (vk;l + vl;k)/2 and σ kl = σ lk. Taking into account the boundary conditions

finally yields

_

V

σ klδdkl dv = _

At

t l

(n)δvl da + _

V

ρf lδvl dv − _

V

ρ

Dvl

Dt

δvl dv. (1.343)

Equation (1.53) expresses the principle of virtual power. Notice that the principle of

virtual work is of no avail here since the expression σ klδul;k cannot be simplified because of

the presence of nonlinear terms in the definition of the Eulerian strain measure. Accordingly,

the spatial description implies a rate formulation and necessitates a thorough discussion of

objective rate tensors. This can be largely avoided by using the material description.

Naturally, the strong form can also be obtained starting from the weak form. Interested

readers are referred to (Belytschko et al. 2000).

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