A Continuum Theory for the Natural Vibrations of Spherical Viral Capsids

Z

s s β ⊗

F (x, t) := α(x, ζ)S(x, ζ; t) B(x, ζ) dζ = αSg dζ e ,

β

I I

Z

Z

s s β (4.2)

⊗

M (x, t) := α(x, ζ)ζS(x, ζ; t) B(x, ζ) dζ = αζSg dζ e ,

β

I I

Z

(3)

f (x, t) := α(x, ζ)S(x, ζ; t)n(x) dζ.

I (3)

s s

We call F the M the and f the

force tensor, moment tensor shear vector.

4.1.2 External Virtual Work

The system of external loads is made of: ni in

• the per unit volume d = d + d , the former addend

distance force o o o

being its part, the latter its part. Denoting the mass

non-inertial inertial

density of the shell by δ , we have

o

in −δ

d (x, ζ; t) = ü(x, ζ; t);

o

o

• the c per unit area, acting on the boundary ∂G of the shell.

contact force o 33

4.1. WEAK FORMULATION

− ±

+

S ∪S S {x ± ∈ S},

We write ∂G = , where := εn(x) : x and use the notation

± ±ε)

Ψ (x) := Ψ(x,

±

S

for the restriction of a field Ψ to .

The expenditure of over the shell-like body is

external virtual work

Z Z

ext

W · ·

(G) [δu] := d δu + c δu.

o o

G ∂G

Again, choosing δu of the same form as (3.1) and following the same procedure

set forth in [7], we get Z

ext

W {q · · · ·

(G) [δa, δw, δϕ, δγ] = δa + (q n)δw + r δϕ + (r n)δγ} ,

o o

o o

S (4.3)

where Z − −

+ +

q (x, t) := α(x, ζ) d (x, ζ; t) dζ + α (x)c (x) + α (x)c (x),

o

o o o

I

Z (4.4)

− −

+ + −

α(x, ζ)ζd (x, ζ; t) dζ + ε α (x)c (x) α (x)c (x)

r (x, t) := o

o o o

I

are the and the per unit area, respectively. The

distance force distance couple

in ni

decomposition d = d + d implies analogous decompositions for q and r :

o o

o

o o

in ni in ni

q = q + q , r = r + r ; in particular, we find that

o

o o o o o  

Z  

(1)

(0)

2 2 2 2

ε ∂ u 2ε ∂ u

in

in −2εδ

α(x, ζ) d (x, ζ; t) dζ = 1+ (x, t) + (x, t) ,

q (x, t) = o

o o  

2 2 2

3ρ ∂t 3ρ ∂t

o

I o

 

Z  

(0) (1)

2

2 2

2 ∂ u 1 ε ∂ u

in 3

in −2ε

α(x, ζ)ζd (x, ζ; t) dζ = δ

r (x, t) = (x, t) + + (x, t) .

o

o o  

2 2 2

3ρ ∂t 3 5ρ ∂t

o

I o (4.5)

34 CHAPTER 4. FIELD EQUATIONS

4.1.3 Principle of Virtual Work

All in all, the weak formulation of the equilibrium problem (Principle of Virtual

) reads:

Work ext int

∀δa, W W

δw, δϕ, δγ, (G) [δa, δw, δϕ, δγ] = (G) [δa, δw, δϕ, δγ] ,

that is to say, Z {q · · · ·

δa + (q n)δw + r δϕ + (r n)δγ} =

o o

o o

S

Z s s s s s T s

· ∇δa · ∇δϕ · ∇δw

F + M + F n + (4.6)

S s T s s s

· ∇δγ · ∇n)

+ M n + ( F δw +

o

(3) s s (3)

· · ∇n ·

+ f n + M δγ + f δϕ .

4.2 Balance Equations

The weak formulation (4.6) can be rewritten, by integration by parts, as follows:

Z n o

s s s s (3)

− · − ·

(− Div F q ) (δa + δwn) + (− Div M + f r ) (δϕ + δγn) = 0,

o

o

S (4.7)

whence, by localization, the following point-wise balance equations have to be

S:

satisfied in s s

Div F + q = 0,

o (4.8)

s s (3)

−

Div M f + r = 0.

o

To provide a component-wise version of these equations, we first define

Z

s (3) i

⊗ ⊗

F := F + f n = αSg dζ e ,

i

I

Z Z (4.9)

s (3) i (3)

⊗ ⊗

M := M + m n = αζSg dζ e , m := αζSn dζ

i

I I 35

4.3. CONSTITUTIVE ASSUMPTIONS αβ α β

· ⊗

We call the contravariant components F := F e e =

membrane forces

s α β

· ⊗

F e e , for α = β, for

normal membrane forces shear membrane forces

αα α α s α α

6 · ⊗ · ⊗

α = β; and we call M := M e e = M e e (α = 1, 2) the bending

αβ α β s α β

· ⊗ · ⊗ 6

and M := M e e = M e e (α, β = 1, 2; α = β) the

moments (3)

3α α

·

Finally, we call F := f e the and

twisting moments. transverse shears

3α (3) α

·

M := m e the thickness moments.

It can be shown that the following system of six scalar equations, involving

the physical components of the tensor fields defined in (4.9), is equivalent to

the vector equations (4.8):

−

(sin ϑ F ) , +F , cos ϑ F + sin ϑ F + ρ sin ϑ q = 0,

<11> <12> <22> <31> <1>

1 2 o o

(sin ϑ F ) , +F , + cos ϑ F + sin ϑ F + ρ sin ϑ q = 0,

<12> <22> <12> <32> <2>

1 2 o o

− −

(sin ϑ F ) , +F , sin ϑ (F + F ρ q ) = 0,

<31> <32> <11> <22> <3>

1 2 o o

− −

(sin ϑ M ) , +M , cos ϑ M ρ sin ϑ F + ρ sin ϑ r = 0,

<11> <12> <22> <31> <1>

1 2 o o o

−

(sin ϑ M ) , +M , + cos ϑ M ρ sin ϑ F + ρ sin ϑ r = 0,

<12> <22> <12> <32> <2>

1 2 o o o

− −

(sin ϑ M ) , +M , sin ϑ (M + M + ρ F ρ r ) = 0.

<31> <32> <11> <22> <33> <3>

1 2 o o o (4.10)

4.3 Constitutive Assumptions

For the shell under study, we consider a linearly elastic response of the following

kind: at any point with respect to the direction of n.

transversely isotropic

Consider the orthonormal basis for the linear space Sym, given by the following

tensors: 1

√ ⊗ ⊗ ⊗

V = (g n + n g ) (β = 1, 2), V = n n,

<β> <β> 3

β 2 1

√

⊗ ⊗ ⊗

W = g g (β not summed), W = (g g + g g ) .

<β> <β> <1> <2> <2> <1>

3

β 2 (4.11)

Given this basis, a representation formula for the elasticity tensor reflecting our

constitutive assumption can be found in [18]; five independent elastic moduli

36 CHAPTER 4. FIELD EQUATIONS

are involved in this representation. However, in technical applications, these

quantities are replaced by six technical moduli – two Young-like, three Poisson-

like and one shear-like – that must satisfy an algebraic condition. The technical

∈ S,

moduli, which we assume to vary with point x enter the representation of

1

the i.e., the inverse of the elasticity tensor as follows :

compliance tensor, C,

1

1

−1 ⊗ ⊗ ⊗

(W W + W W ) + V V +

=

C 1 1 2 2 3 3

E E

p n

ν ν

p pn

− ⊗ ⊗ − ⊗ ⊗

(W W + W W ) (V W + V W ) +

1 2 2 1 3 1 3 2

E E

p p (4.12)

ν 1

np

− ⊗ ⊗ ⊗ ⊗

(W V + W V ) + (V V + V V ) +

1 3 2 3 1 1 2 2

E 2G

n

1 + ν

p ⊗

+ W W .

3 3

E

p

Due to the built-in symmetries of it results that

C, ν

E pn

p = .

E ν

n np

In order to understand the mechanical meaning of the technical moduli, we

·

preliminarily fix an arbitrary unit vector e such that e n = 0, and an arbitrary

· ·

unit vector s such that s n = 0 and s e = 0.

First, consider an uniaxial stress in the direction e induced in a specimen

⊗

made of the material under examination: S = σe e. Then, the corresponding

−1

strain is E = [S] and we find that the ratio between the axial stress and

C

the axial strain is the in-plane Young’s modulus

· ⊗

S e e

E = ;

p · ⊗

E e e

moreover, we find the and respec-

in-plane first transverse Poisson’s moduli,

tively given by · ⊗ · ⊗

E s s E n n

− −

ν = , ν = .

p pn

· ⊗ · ⊗

E e e E e e

1 Here and in the sequel, we will left tacit the dependence on x of the elasticity and com-

pliance tensors, as well as of the technical moduli. 37

4.3. CONSTITUTIVE ASSUMPTIONS ⊗

Now, consider an uniaxial stress in the direction n, S = σn n. Analogously,

given the corresponding strain E, we find the and

transverse Young’s modulus

the second transverse Poisson’s modulus:

· ⊗ · ⊗

S n n E e e

−

E = , ν = .

n np

· ⊗ · ⊗

E n n E n n

⊗ ⊗

Finally, consider a shear stress of the form S = τ (e n + n e) and the

corresponding strain E. Then, we find the given by

transverse shear modulus,

· ⊗

S e n

2G = .

· ⊗

E e n

The elasticity tensor can be represented, in terms of the technical moduli, by

inversion of (4.12); it results

E

p − ⊗ ⊗

[(1 ν ν ) (W W + W W ) +

=

C pn np 1 1 2 2

∆ ⊗ ⊗

+ (ν + ν ν ) (W W + W W ) +

p pn np 1 2 2 1

⊗ ⊗

+ (1 + ν )ν W V + (1 + ν )ν W W ] +

p np 1 3 p pn 2 3 (4.13)

E

n ⊗ ⊗

+ [(1 + ν )ν (V W + V W ) +

p pn 3 1 3 2

∆

2

− ⊗ ⊗ ⊗

+ (1 ν )V V + 2G (V V + V V ) +

3 3 1 1 2 2

p

E

p ⊗

+ W W ,

3 3

1 + ν

p

where − −

∆ := (1 + ν )(1 ν 2ν ν ).

p p pn np

38 CHAPTER 4. FIELD EQUATIONS

Component-wise, the constitutive law S = w