[PPT] - GENERALIZED STRESS CONCENTRATION FACTORS Reuven Segev Department of PowerPoint Presentation

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GENERALIZED STRESS CONCENTRATION FACTORS

Reuven Segev

Department of Mechanical Engineering Ben-Gurion University, Beer-Sheva, Israel

ICTAM2004, Warsaw

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Stress Concentration for Engineers

R. Segev

ICTAM2004, Warsaw

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Generalized Stress Stress Concentration Factors:

Assume a body Ω is given (open, regular with smooth boundary).
Assume a force F is given in terms of a body force b and a surface

force t and let σ be a stress field that is in equilibrium with F.

The stress concentration factor associated with the pair F, σ is

KF,σ = ess supx {|σ(x)|} ess supx,y {|b(x)| , |t(y)|} , x ∈ Ω, y ∈ ∂Ω.

Denote by F the collection of all possible stress fields that are in

equilibrium with F. (There are many such stress fields because material properties are not specified.)

R. Segev

ICTAM2004, Warsaw

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The optimal stress concentration factor for the force F is defined by

KF = inf

σ∈F

KF,σ
.
The generalized stress concentration factor K—a purely geometric

property of Ω—is defined by K = sup

F

{KF} = sup

F

inf

σ∈F

ess supx {|σ(x)|}

ess supx,y {|b(x)| , |t(y)|}

.

Result: K = δ , where, δ is a mapping that extends vector fields from the interior

f the body to its closure and is defined on Sobolev spaces or the

related L D-spaces.

R. Segev

ICTAM2004, Warsaw

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First case: General mechanics – simpler mathematics

Forces may have non-vanishing resultants and total torques.
The stress object contains a tensor field σi j and a self force field σ0i.
The principle of virtual work is of the form
Ω

biwidV +

∂Ω

tiwidA =

Ω

σ0iwidV +

Ω

σi jwi, jdV.

The stress tensor σi j need not be symmetric.
With the self force field

K = sup

F

inf

σ∈F

ess supi, j,k,x

|σ0i(x)| ,

σ jk(x)
ess supi,x {|bi(x)| , |ti(x)|}
.
R. Segev

ICTAM2004, Warsaw

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Forces and Stresses as Linear Functionals

A Force: A linear functional (power functional) on virtual velocity fields, F(w) =

Ω

biwidV +

∂Ω

tiwidA. A Stress: A linear functional (power functional) on the space of tensor fields (gradients of virtual velocity fields) σ(χ) =

Ω

σi jχi jdV. We will generalize stresses to include self-forces so σ(χ) =

Ω

σ0iχidV +

Ω

σi jχi jdV. Equilibrium: F(w) = σ(χ) if χi = wi and χi j = wi, j.

R. Segev

ICTAM2004, Warsaw

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The L1 and L∞-Norms and Their Duality

Objective: The maximal absolute value of a stress component will be the magnitude or norm of the stress σL∞ = ess sup

i, j,k,x

|σ0i(x)| ,

σ jk(x)
.

Duality: If we use the L1-norm on the space of “local deformations” {χ = (χi, χ jk)}, χL1 =

i
Ω

|χi| dV +

i, j
Ω
χi j
dV,

then every stress with finite L∞-norm is continuous and σL∞ = σL1∗ = sup

χ

|σ(χ)| χL1 = sup

χL1=1

|σ(χ)| .

R. Segev

ICTAM2004, Warsaw

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The Measure µ and the Corresponding Norms

Objective: Set F = max

ess sup

i,x

{|bi(x)|} , ess sup

j,y

{|t j(y)|}

.

This will be the dual norm of a force if we use the norm wL1,µ =

i
Ω

|wi| dV +

i
∂Ω

|wi| dA = wL1 + w|∂ΩL1 . This is the L1-norm relative to the measure µ, denoted L1,µ, such that µ(D) = V(D ∩ Ω) + A(D ∩ ∂Ω). Conclusion: We want to find a relation between the L∞-norm of the stress field and the L∞,µ-norm F of the force.

R. Segev

ICTAM2004, Warsaw

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The Relation Between L1,µ(Ω, R3) and L1(Ω, R12)

L1,µ(Ω, R3)

δ

← − − − −

j

− − − − → L1(Ω, R12) L1,µ(Ω, R3)∗

δ∗

− − − − →

∗ j∗

← − − − − L1(Ω, R12)∗

L∞,µ(Ω, R3)

L∞(Ω, R12)

R. Segev

ICTAM2004, Warsaw

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Sobolev mappings, traces and extensions

We consider the Sobolev space W 1

1 (Ω, R3) of L1-mappings whose

distributional derivatives are also L1-mappings. The Sobolev space is a Banach space under the norm φW 1

1 =

i

φiL1 +

j,k
φ j,k
L1

.

There is a continuous linear mapping, the trace mapping

γ : W 1

1 (Ω, R3) −

→ L1(∂Ω, R3), γ (u

∂Ω)(y) = u(y), y ∈ ∂Ω, u ∈ C(Ω, R3).

Thus, there is a K∂ > 0 such that γ (w)L1 K∂ wW 1

1 .

R. Segev

ICTAM2004, Warsaw

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Implications to the Present Situation

Clearly, we have a continuous inclusion mapping

ι0 : W 1

1 (Ω, R3) −

→ L1(Ω, R3), satisfying ι0(w)L1 wW 1

1 .

Hence, there is a linear injection—the extension to the boundary—

δ : W 1

1 (Ω, R3) −

→ L1,µ(Ω, R3), δ(w)(x) = w(x), x ∈ Ω, δ(w)(y) = γ (w)(y), y ∈ ∂Ω.

The extension to the boundary is continuous and its norm is

δ = sup

w

δ(w)L1,µ wW 1

1

= sup

w∈W 1

1 (Ω,R3)

Ω

|w| dV +

∂Ω
ˆ

w

dA
Ω

|w| dV +

Ω

|∇w| dV .

R. Segev

ICTAM2004, Warsaw

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The Relation Between the L∞,µ and W 1

1 ∗-Norms of Forces

As δ is a linear continuous injection, the dual mapping

δ∗ : L1,µ(Ω, R3)∗ = L∞,µ(Ω, R3) − → W 1

1 (Ω, R3)∗,

δ∗(F)(w) = F(δ(w)), for all w ∈ W 1

1 (Ω, R3), is continuous.

A basic implication of the Hahn-Banach theorem: δ∗ = δ.

Thus, sup

F

δ∗(F)W 1

1 ∗

FL∞,µ = δ , F ∈ L∞,µ(Ω, R3).

R. Segev

ICTAM2004, Warsaw

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The Representation of W 1

1 -Forces by Stresses in L∞(Ω, R12)

Consider the injection

j : W 1

1 (Ω, R3) −

→ L1(Ω, R12), j(φ) = (φi, φl,m).

We note that φW 1

1 = j(φ)L1.

It follows that every W 1

1 -force S may be represented (non-uniquely)

by some stress σ in L∞(Ω, R12) in the form S = j∗(σ).

In addition,

SW 1

1 ∗

= inf

S= j∗(σ) σL∞ .

R. Segev

ICTAM2004, Warsaw

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Result

The situation so far L1,µ(Ω, R3)

δ

← − − − − W 1

1 (Ω, R3) j

− − − − → L1(Ω, R12) L1,µ(Ω, R3)∗

δ∗

− − − − → W 1

1 (Ω, R3)∗ j∗

← − − − − L1(Ω, R12)∗

L∞,µ(Ω, R3)

L∞(Ω, R12)

R. Segev

ICTAM2004, Warsaw

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✬ ✫ ✩ ✪ δ = δ∗ = sup

F∈L∞,µ(Ω,R3)

δ∗(F)W 1

1 ∗

FL∞,µ = sup

F∈L∞,µ(Ω,R3)

infσ, δ∗(F)= j∗(σ)

ess supi, j,k,x

|σ0i(x)| ,

σ jk(x)
ess supi,x {|bi(x)| , |ti(x)|}

We recall that δ∗(F) = j∗(σ) means δ∗(F)(w) = j∗(σ)(w). It follows that σ ∈ F because

Ω

biwidV +

∂Ω

tiwidA =

Ω

σ0iwidV +

Ω

σi jwi, jdV. Hence, δ = K

R. Segev

ICTAM2004, Warsaw

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Adaptation to Equilibrated Forces and Stresses

Basic idea: consider forces in the various dual spaces that do not perform power on rigid velocity fields (in- finitesimal displacement fields).

R. Segev

ICTAM2004, Warsaw

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Stretchings and Rigid Velocity Fields

For a velocity field w ∈ W, the associated stretching (strain,

deformation) ε(w) is the tensor field ε(w)im = 1 2(wi,m + wm,i).

A rigid velocity (or displacement) field is of the form

w(x) = a + ω × x, x ∈ Ω.

R – the collection of rigid velocity fields—a 6-dimensional subspace
f the spaces of velocity fields.
Considering the kernel of the stretching mapping ε: w → ε(w), a

theorem whose classical version is due to Liouville states that Kernel ε = R. In particular, ε(w + r) = ε(w).

R. Segev

ICTAM2004, Warsaw

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Distortions and Approximations by Rigid Velocities

For a space W of velocity fields, the associated space of distortions is

W /

R. On W

/ R the stretching map ε/ R([w]) is well defined. We have the natural projection π : W → W / R.

A norm is induced on W

/ R by [w] = infr∈R w − r – the error

f the best approximation by rigid motion.
For the L2-norm, the best approximation is the rigid motion that

gives the same linear and angular momentum as w. This gives a projection πR : W → R. – Setting W0 = Kernel πR ⊂ W, we have an isomorphism W / R ∼ = W0 and 0 − − − − → R

ιR

− − − − → W

π

− − − − → W / R − − − − → 0 0 ← − − − − R

πR

← − − − − W

ι

← − − − − W0 ← − − − − 0. – Thus, W ∼ = W0 ⊕ R.

R. Segev

ICTAM2004, Warsaw

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Equilibrated Forces

W – a vector space of velocities (we assume that it contains the rigid

velocities).

A force F ∈ W∗ is equilibrated if F(r) = 0 for all r ∈ R.
As the quotient projection is surjective, the dual mapping

π∗ : (W / R)∗ → W∗ is injective and its image is the collection of equilibrated forces. Equilibrated forces ∼ = (W / R)∗

π∗ is norm preserving. Thus, we may identify the collection of

equilibrated forces in W∗ with (W / R)∗. 0 − − − − → R

ιR

− − − − → W

π

− − − − → W / R − − − − → 0 0 ← − − − − R∗

ι∗

R

← − − − − W∗

π∗

← − − − − (W / R)∗ ← − − − − 0.

Using the projection πR and the Whitney sum structure it induces we

have a Whitney sum structure W∗ = W∗

0 ⊕ R∗.

R. Segev

ICTAM2004, Warsaw

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General Setting

We had: Velocities: L1,µ(Ω, R3)

δ

← − − − − W 1

1 (Ω, R3) j

− − − − → L1(Ω, R12) Forces: L1,µ(Ω, R3)∗

δ∗

− − − − → W 1

1 (Ω, R3)∗ j∗

← − − − − L1(Ω, R12)∗ Replace spaces of velocities W by the corresponding spaces of distortions W / R and replace the spaces of forces W∗ by the corresponding (W / R)∗. L1,µ(Ω, R3) / R

δ/

R

← − − − − ? / R

ε/

R

− − − − → L1(Ω, R6) (L1,µ(Ω, R3) / R)∗

(δ/

R)∗

− − − − →

? /

R ∗

(ε/

R)∗

← − − − − L∞(Ω, R6) ? should satisfy the following conditions:

ε/

R should be a norm preserving injection.

δ and δ/

R should be well defined and in particular, the trace theorem should hold.

R. Segev

ICTAM2004, Warsaw

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✬ ✫ ✩ ✪ Ornstein, Arch. Rat. Mech. Anal.(1962) W 1

1 (Ω, R3) is not suitable.

Strauss(1973), Temam & Strang(1980), Kohn(1982), Temam(1983) ? = L D(Ω) has the required properties. L D(Ω) – the space of integrable stretchings (deformations, strains): w = πR(w) + ε(w)L1 .

R. Segev

ICTAM2004, Warsaw

Bad News Good News

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Properties of L D(Ω)

Definition: The collection of fields for which wL D =

i

wiL1 +

i,m

ε(w)imL1 is finite and serves as a norm, L D(Ω) is a Banach space. Approximation: C∞(Ω, R3) is dense in L D(Ω). Traces: There is a unique continuous linear mapping γ : L D(Ω) − → L1(∂Ω, R3) such that γ (u

Ω) = u
∂Ω, u ∈ C(Ω, R3).

Extension to the boundary: There is a unique continuous extension to the boundary δ : L D(Ω) − → L1,µ(Ω, R3), such that δ(u

Ω) = u, for every u ∈ C(Ω, R3).
R. Segev

ICTAM2004, Warsaw

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✬ ✫ ✩ ✪ Distortions of integrable stretching: On the space of L D-distortions, L D(Ω) / R, the norm χ = inf

w∈χ wL D is equivalent to ε/

R([w])L1 =

i,m

ε(w)imL1 . Equivalent norms: wL D =

i

wiL1 +

i,m

ε(w)imL1 is equivalent to πR(w) + ε(w)L1 = πR(w) + ε/ R([w])L1 .

R. Segev

ICTAM2004, Warsaw

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Resulting Structure

L1,µ(Ω, R3)

δ

← − − − − L D(Ω)

ε

− − − − → L1(Ω, R6)

π

 



 π

L1,µ(Ω, R3)

/ R

δ/

R

← − − − − L D(Ω) / R

ε/

R

− − − − → L1(Ω, R6) L∞,µ(Ω, R3)

δ∗

− − − − → L D(Ω)∗

ε∗

← − − − − L∞(Ω, R6)

π∗







π∗

(L1,µ(Ω, R3)

/ R)∗

(δ/

R)∗

− − − − → (L D(Ω) / R)∗

(ε/

R)∗

← − − − − L∞(Ω, R6) K = δ/ R

R. Segev

ICTAM2004, Warsaw

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Using (π0, πR): L D(Ω) ⇐ ⇒ L D(Ω)0 ⊕ R

L1,µ(Ω, R3)

δ

← − − − − L D(Ω)

ε

− − − − → L1(Ω, R6)

π0

 



 π0

L1,µ(Ω, R3)0

δ0

← − − − − L D(Ω)0∼ =L D(Ω) / R

ε

− − − − → L1(Ω, R6) with quotient norm isometric (L1,µ(Ω, R3)∗

δ∗

− − − − → L D(Ω)∗

0∼

= (L D(Ω) / R)∗

ε∗

← − − − − L∞(Ω, R6) K = δ0

R. Segev

ICTAM2004, Warsaw