Research Activities and Scientiﬁc Results of Mircea BÎRSAN · 2015-08-19 · Research Activities...

Research Activities Scientific Results Future Plans

Research Activities and Scientific Resultsof Mircea BÎRSAN

Faculty of Mathematics, University “A. I. Cuza” of Iasiand

Fakultät für Mathematik, Universität Duisburg-Essen

– Habilitation Thesis –


RESEARCH ACTIVITIES


Research Interests

1997 – 2003 : Doctoral studies under the supervision ofProfessor Dorin Iesan

July 4, 2003 : PhD–Thesis entitled Study of thedeformation of some elastic solids with microstructure

My research activity focusses on the mathematicalproblems of the theory of elasticity and thermoelasticity

The theory of shells and plates is an important chapter ofthe elasticity theory

The shell theory has important applications in engineering:renewed interest for the derivation of refined shell theoriesand for the mathematical justification of the existing models


2003 – 2006

Working directions :

I have investigated the equations of Mindlin-type plates

I have extended the classical model to the case ofthermoelastic porous materialsI have studied the governing equations:

existence and uniqueness of solutioncontinuous dependence of solution upon body loads andinitial datareciprocity, variational theoremsrepresentation of the solution, oscillations etc.

I have published my first book:M. Birsan : Deformation of elastic porous plates: Amathematical study, Ed. Matrix Rom, Bucuresti, 2007.


Selected publications on this subject :

• M. Birsan (2007): On the bending equations for elastic plateswith voids, Mathematics and Mechanics of Solids, 12, 40–57

• M. Birsan (2006): Transient and steady-state solutions forporous thermoelastic plates, Analele Stiintifice Univ. Iasi, Ser.Matematica, 52, 159–176

• M. Birsan (2004): Some theorems in the bending theory ofporous thermoelastic plates, Analele Stiintifice Univ. Iasi, Ser.Matematica, 50, 305–314

• M. Birsan (2003): A bending theory of porous thermoelasticplates, Journal of Thermal Stresses, 26, 67–90


2004 – 2007I have investigated the theory of Cosserat surfaces:deformable surfaces with a single deformable directorattached to each pointI have extended the existing theory to some microstructuralmaterials, such as porous thermoelastic materialsI have proved a Korn-type inequality for this model andgeneral theorems concerning the governing field equationsI have determined analytical solutions for cylindrical shells,which are the counterpart of Saint-Venant’s solutions in theclassical elasticity theoryI have solved analytically some thermal stresses problemsfor cylindrical shells, which are of practical interest2007: “Gheorghe Lazar” - Prize of the Romanian AcademyI have published my second book:M. Birsan : Linear Cosserat Elastic Shells: MathematicalTheory and Applications, Ed. Matrix Rom, Bucuresti, 2009.



• M. Birsan (2008): Inequalities of Korn’s type and existenceresults in the theory of Cosserat elastic shells,Journal of Elasticity, 90, 227-239.

• M. Birsan (2005): Minimum energy characterizations for thesolution of Saint-Venant’s problem in the theory of shells,Journal of Elasticity, 81, 179-204.

• M. Birsan (2004): 16) M. Birsan (2004): The solution ofSaint-Venant’s problem in the theory of Cosserat shells,Journal of Elasticity, 74, 185-214.

• M. Birsan (2009): Thermal stresses in cylindrical Cosseratelastic shells, European J. Mechanics A/Solids, 28, 94-101

• M. Birsan (2009): On Saint-Venant’s problem for anisotropic,inhomogeneous, cylindrical Cosserat elastic shells,International Journal of Engineering Science, 47, 21-38


• M. Birsan (2007): On the theory of loaded general cylindricalCosserat elastic shells, International Journal of Solids andStructures, 44, 7399-7419.

• M. Birsan (2007): On Saint-Venant’s principle in the theory ofCosserat elastic shells, International Journal of EngineeringScience, 45, 187-198.

• M. Birsan (2006): On a thermodynamic theory of porousCosserat elastic shells, Journal of Thermal Stresses, 29,879-899.

• M. Birsan (2006): On the theory of elastic shells made from amaterial with voids, International Journal of Solids andStructures, 43, 3106-3123.

• M. Birsan (2006): Several results in the dynamic theory ofthermoelastic Cosserat shells with voids,Mechanics Research Communications, 33, 157-176.


2008 – 2011

I received a Humboldt Research Fellowship at theUniversity of Halle-Wittenberg (Germany); my host wasProf. Holm Altenbach

I focussed my attention to the model of directed shells,which considers every point of the deformable surface asan infinitesimal rigid body: this approach can beconveniently applied to multi-layered shells.

we have performed a mathematical study of the dynamicalequations of thermoelastic orthotropic directed shells

we have solved the equilibrium equations to determine thebending, extension, torsion and flexure deformation forcylindrical multi-layered shells

the characterization of the effective stiffness properties oflayered thermoelastic shells


Another topic of interest:

the theory of Cosserat curved rods : every point of thedeformable curve is viewed as an infinitesimal rigid body

this approach is able to describe the effective mechanicalproperties of composite rods or porous thermoelastic rods:we have extended the model in these directions

E.U. 7th Framework Programme research project (2011):Modern Composites Applied in Aerospace and SurfaceTransport Infrastructure , at University of Technology inLublin (Poland)

the technical facilities available in this research projectallowed us to compare the theoretical/mathematical resultswith experimental measurements and numerical simulation

2012: “Nicolae Teodorescu” - Prize of the Academy ofRomanian Scientists for my papers in 2010–2011.



• M. Birsan, H. Altenbach (2011): On the dynamical theory ofthermoelastic simple shells, ZAMM – Journal of AppliedMathematics and Mechanics, 91, 443-457.

• M. Birsan, H. Altenbach (2010): A mathematical study of thelinear theory for orthotropic elastic simple shells,Mathematical Methods in Applied Sciences, 33, 1399-1413.

• H. Altenbach, M. Birsan, V.A. Eremeyev (2012): On athermodynamic theory of rods with two temperature fields,Acta Mechanica, 223, 1583–1596.

• M. Birsan, H. Altenbach (2012): The Korn-type inequality in aCosserat model for thin thermoelastic porous rods,Meccanica, 47, 789–794.

• M. Birsan, H. Altenbach (2011): On the theory of porouselastic rods, International J. Solids and Structures, 48, 910-924.


• M. Birsan, H. Altenbach (2011): Theory of thin thermoelasticrods made of porous materials, Archive of Applied Mechanics,81, 1365–1391.

• M. Birsan, T. Sadowski, L. Marsavina, E. Linul, D. Pietras(2013): Mechanical behavior of sandwich composite beamsmade of foams and functionally graded materials,International Journal of Solids and Structures, 50, 519–530.

• M. Birsan, T. Sadowski, D. Pietras (2013): Thermoelasticdeformations of cylindrical multi-layered shells using a directapproach, Journal of Thermal Stresses, vol. 36, in print.

• M. Birsan, H. Altenbach, T. Sadowski, V. Eremeyev, D. Pietras(2012): Deformation analysis of functionally graded beams bythe direct approach, Composites Part B: Engineering, 43,1315–1328.


2011 – 2013

I started the collaboration with Prof. Patrizio Neff at theFaculty of Mathematics, University Duisburg - Essen(Germany)

we focussed our attention on the non-linear theory of6-parameter shells : this is a general approach which takesadditionally into account the drilling rotations of the points

we have proved the existence of minimizers forgeometrically non-linear 6-parameter plates or shells, forisotropic and anisotropic materials

we are working on our own model of curved shells,obtained by dimensional reduction from a 3D Cosseratbody, which is convenient for numerical implementation

we intend to show that it is mathematically well-posed andthen to extend it to the case of elasto–plastic shells



• M. Birsan, P. Neff (2013): Existence theorems in thegeometrically non-linear 6-parameter theory of elastic plates,Journal of Elasticity, DOI: 10.1007/s10659-012-9405-2 .

• M. Birsan, P. Neff (2013): Existence of minimizers in thegeometrically non-linear 6-parameter resultant shell theory withdrilling rotations, Mathematics and Mechanics of Solids,published online first DOI: 10.1177/1081286512466659 .

• M. Birsan, P. Neff (2012): On the equations of geometricallynonlinear elastic plates with rotational degrees of freedom,Ann. Acad. Rom. Sci. Ser. Math. Appl., vol. 4, no. 1, 97-103.

• M. Birsan, P. Neff (2013): On the characterization of drillingrotation in the 6-parameter resultant shell theory. In: W.Pietraszkiewicz, J. Górski (eds.), Shell Structures: Theory andApplications, vol. 3, Taylor & Francis, in print.


Cosserat Curved Rods and Beams


Kinematical model of directed curves• Proposed by ZHILIN (2006, 2007): the rod model consists in adeformable curve with a triad of rigidly rotating orthonormalvectors connected to each point.

O

r

C0

d3 = t

d1d2


Features of the direct approach :

• No need for mathematical manipulations with 3D equations.

• The basic laws of mechanics are applied directly to aone-dimensional continuum.

• To formulate the constitutive equations, we have to determinethe structure of the elasticity tensors and to identify the effectiveproperties: we make use of the effective stiffness concept.

The reference configuration C0 of the rod is given by the vectorfields:

r(s), di(s), i = 1,2,3,

where s is the arclength and di are the directors.


R(s, t)

u(s, t)

nb

σ

D3

D1

D2

O

r(s)C0

d3

d1

d2

Reference and deformed configuration of the rod


The motion of the rod is defined by the functions

R = R(s, t), Di = Di(s, t), i = 1,2,3, s∈ [0, l].

The displacement vector : u(s, t) = R(s, t) − r(s),

and the rotation tensor : P(s, t) = Dk(s, t)⊗ dk(s).

We denote by :

V the velocity vector : V(s, t) = R(s, t) ;

ω the angular velocity : P(s, t) = ω(s, t) × P(s, t).

We have ω = axial(

P · PT)

= −12

[

P · PT]

×.


Porosity

We use the Nunziato–Cowin theory for elastic materials withvoids (1979, 1983).

The mass density of the porous rod ρ = ρ(s, t) is representedas the product :

ρ(s, t) = ν(s, t) γ(s, t) ,

where γ(s, t) is the mass density of the matrix elastic material.

The porosity variable is:

the volume fraction field : ν = ν(s, t).

The field ν(s, t) describes the continuous distribution of voidsalong the rod. (0< ν ≤ 1)


Temperature

The absolute temperature in the rod is :

θ = θ(s, t) > 0.

The basic laws of thermodynamics are applied directly to thedeformable curve.

For instance, the Clausius-Duhem inequality for the entropy ofthe rod is

∫ s2

s1

ρ0 η ds ≥∫ s2

s1

ρ0Sθ

ds +(qθ

)∣

∣

∣

s2

s1

, ∀ s1, s2 ∈ [0, l].


Equations of motion

Equation of linear momentum:

N ′(s, t) + ρ0F = ρ0ddt

(V +Θ1 · ω).

Equation of moment of momentum :

M ′(s, t) + R ′ × N(s, t) + ρ0L =

= ρ0[

V ×Θ1 · ω +ddt

(V ·Θ1 +Θ2 · ω)]

.

Equation of equilibrated force :

h ′(s, t)− g(s, t) + ρ0 p = ρ0ddt

(κ ν) .


Energy balance

Equation of energy balance :

ρ0U = P + ρ0S+ q ′ ,

with P = N·(V′+R′×ω) + M ·ω′ + gν + hν ′.

Entropy inequality :

ρ0 θ η ≥ ρ0S + q ′ − θ′

θq .

Introduce the Helmholtz energy function :

Ψ = U − θ η .


The expression of the energy function Ψ :

ρ0Ψ = Ψ0 + N0 · E + M0 ·Φ+12E · A · E

+E · B ·Φ+12Φ · C ·Φ+Φ · (E · D) ·Φ

+12

K1 ν2 +

12

K2(ν′)2 + K3 ν ν

′ + (K4 · E) ν

+(K5 ·Φ) ν + (K6 · E) ν ′ + (K7 ·Φ) ν ′

−(G1 · E) θ − (G2 ·Φ) θ − G3 ν θ − G4 ν′ θ − 1

2Gθ2,

where E is the vector of extension–shearand Φ is the vector of bending–twisting.

The elasticity tensors A, B, C and D are known (Zhilin, 2006)

We have determined the structure of the tensors K1 , ... , K7

and G1,..., G4 , describing the poro-thermoelastic properties.


Linear theoryWe denote by ψ(s, t) the vector of infinitesimal rotations andT and ϕ the variations of the temperature and porosity fields.The constitutive equations are:

N =∂(ρ0Ψ)

∂e, M =

∂(ρ0Ψ)

∂κ,

η = −∂Ψ∂T

, g =∂(ρ0Ψ)

∂ ϕ, h =

∂(ρ0Ψ)

∂(ϕ ′),

where e and κ are the vectors of deformation.

The equations of motion become :

N′ + ρ0F = ρ0 (u +Θ01 · ψ),

M′ + t × N + ρ0L = ρ0(

u ·Θ01 +Θ

02 · ψ

)

,

h′ − g+ ρ0 p = ρ0 κ ϕ .


The reduced energy balance equation is:

(K T ′) ′ + ρ0 S = ρ0 θ0 η ,

where K ≥ 0 is the thermal conductivity of the rod.

The initial-boundary-value problem: adjoin boundary conditions

u(s, t) = u(t) or N(s, t) = N(t),ψ(s, t) = ψ(t) or M(s, t) = M(t),ϕ(s, t) = ϕ(t) or h(s, t) = h(t),T(s, t) = T(t) or q(s, t) = q(t), for s∈ {0, l}.

and appropriate initial conditions. We prove :

Theorem

(Uniqueness) Assume that the mass density ρ0 , the inertiacoefficient κ and the coefficient G are positive.Then the initial–boundary–value problem for porousthermoelastic rods has at most one solution.


Korn Inequality and Existence results

Theorem

(Korn-Type Inequality) For every y =(

ui(s), ψi(s))

we definethe components of the deformation vectors ei(y) and κi(y) inthe Frenet vector basis { t , n , b } .Then, there exists a constant c1 > 0 such that

∫

C

[

ei(y)ei(y) + κi(y)κi(y)]

ds ≥

≥ c1

∫

C

(

ui ui + ψi ψi + u′i u′i + ψ′i ψ

′i

)

ds, ∀ y ∈ V,

where

V ={(

ui , ψi)

∈ H1[0, l] | ui = 0 onΓu , ψi = 0 onΓψ}

.


The inequality of Korn–type can be used to proveexistence results for the equations of porous thermoelastic rodswritten in a weak variational form :

Equilibrium equations: we employ the Lax–Milgram lemma

Dynamical equations: we employ the semigroup of linearoperators theory

These results show that the mathematical theory ofthermoelastic porous curved rods is well–formulated.


Basic equations for straight composite rods(purely elastic case)

Equations of equilibrium :

N ′(s, t) + ρ0F = 0,

M ′(s, t) + R ′ × N(s, t) + ρ0L = 0,

with:

the force vector : N = F e3 + Q1e1 + Q2e2 ;

and the moment vector : M = H e3 + e3 × (L1e1 + L2e2) ;

the displacement vector : u = ue3 + w1e1 + w2e2 ;

and the rotation vector : ψ = ψ e3 + e3 × (ϑ1e1 + ϑ2e2) .


The constitutive equations for elastic directed curves:

Q1 = A1(w′1−ϑ1) + A12(w′

2−ϑ2) + B13ψ′,

Q2 = A12(w′1−ϑ1) + A2(w′

2−ϑ2) + B23ψ′,

F = A3 u′ − B31ϑ′2 + B32ϑ

′1,

H = C3ψ′ + B13(w′

1−ϑ1) + B23(w′2−ϑ2),

L1 = C2ϑ′1 − C12ϑ

′2 + B32 u′,

L2 = −C12ϑ′1 + C1ϑ

′2 − B31 u′ .

We determine the effective stiffness properties

Ai , Ci , A12 , C12 , B13 , B23 , B31 , B32

for various types of composite beams, expressed in terms ofthe 3D elasticity constants of the constituents.


We consider composite beams with general cross-sections,made of an arbitrary number of ‘layers’ and ‘fibers’ :

S1

S2

S3Sm

Sm+1Sm+2

Sm+n

We denote the jump on Ck = ∂Sk by[

f]+

−

= f (r) − f (l) for any field f ,

where f (r) designates the field f calculated in Sr .


Effective stiffness coefficients are determined by comparison ofanalytical solutions for bending, torsion, extension, andvibration problems in the two approaches.

We denote by u(s)α (x1, x2) the solutions of the boundary-valueproblems P(s) , s= 1,2,3, on the cross-section domain Σ:

P(γ) :

tβα, β = −(λ xγ), α in Sk (k = 1, ..., n),tβαnβ = −λ xγnα on ∂Σ, (γ = 1, 2)[

uα]+

−

= 0, nβ[

tβα + λxγδαβ]+

−

= 0 on Ck (k = 1, ..., n),

P(3) :

tβα, β = −λ, α in Sk (k = 1, ..., n),tβαnβ = −λnα on ∂Σ,[

uα]+

−

= 0, nβ[

tβα + λ δαβ]+

−

= 0 on Ck (k = 1, ..., n),


For beams made of isotropic materials we obtain:• extensional stiffness and bending stiffness coefficients

A3 =

n∑

k=1

∫

Sk

(

λ(k) + 2µ(k) + λ(k) u(3)γ,γ)

dx1dx2 ,

C1 =

n∑

k=1

∫

Sk

x2[

(λ(k) + 2µ(k))x2 + λ(k)u(2)γ,γ]

dx1dx2,

C2 =

n∑

k=1

∫

Sk

x1[

(λ(k) + 2µ(k))x1 + λ(k)u(1)γ,γ]

dx1dx2 ;

• effective shear stiffness coefficients

Aα =κArea(Σ)

〈 x2α 〉 ·

n∑

k=1

∫

Sk

µ(k)dx1dx2 ·n

∑

k=1

∫

Sk

ρ(k)x2αdx1dx2

n∑

k=1

∫

Sk

ρ(k)dx1dx2

, (κ = π2

12).


• torsional rigidity

C3 =n

∑

k=1

∫

Sk

µ(k)[

x1(x1 + ϕ,2) + x2(x2 − ϕ,1)]

dx1dx2 ,

• effective coupling coefficients

B31 =

n∑

k=1

∫

Sk

x2(

λ(k) + 2µ(k) + λ(k)u(3)γ,γ)

dx1dx2,

B32 = −n

∑

k=1

∫

Sk

x1(

λ(k) + 2µ(k) + λ(k)u(3)γ,γ)

dx1dx2,

C12 = −n

∑

k=1

∫

Sk

x1[

(λ(k) + 2µ(k))x2 + λ(k)u(2)γ,γ]

dx1dx2 .

Remarks:

If the Poisson ratio is constant, then these formulas simplify

We have determined the effective stiffness coefficients also forporous thermoelastic rods.


For beams made of orthotropic materials (non-homogeneous):

A3 =

n∑

k=1

∫

Sk

(

c(k)33 + c(k)13 u(3)1,1 + c(k)23 u(3)2,2

)

dx1dx2 ,

B31 =n

∑

k=1

∫

Sk

x2(

c(k)33 + c(k)13 u(3)1,1 + c(k)23 u(3)2,2

)

dx1dx2 ,

B32 = −n

∑

k=1

∫

Sk

x1(

c(k)33 + c(k)13 u(3)1,1 + c(k)23 u(3)2,2

)

dx1dx2 ,

C1 =

n∑

k=1

∫

Sk

x2(

c(k)33 x2 + c(k)13 u(2)1,1 + c(k)23 u(2)2,2

)

dx1dx2,

C2 =

n∑

k=1

∫

Sk

x1(c(k)33 x1 + c(k)13 u(1)1,1 + c(k)23 u(1)2,2)dx1dx2 ,

C12 = −n

∑

k=1

∫

Sk

x1(

c(k)33 x2 + c(k)13 u(2)1,1 + c(k)23 u(2)2,2

)

dx1dx2 ,

C3 =

n∑

k=1

∫

Sk

[

c(k)44 x1(x1 + ϕ,2) + c(k)55 x2(x2 − ϕ,1)]

dx1dx2 ,

A1 = κ 〈 c55 〉〈 ρ x2

1 〉Area(Σ)〈 ρ 〉〈 x2

1 〉, A2 = κ 〈 c44 〉

〈 ρ x22 〉Area(Σ)〈 ρ 〉〈 x2

2 〉


Rectangular functionally graded (FGM) beam

Density distribution function – power law (N = 1,2, ...,10)

ρ(x1) = ρm + (ρs − ρm)

(

2|x1|h

)N

.

Young’s modulus for closed-cell aluminum foams (Gibson 1997)

E(x1) = Es

(

ρ(x1)

ρmax

)2

.


The effective stiffness coefficients are given by:

A3 = bh Es

[

r2 +2r(1− r)

N + 1+

(1− r)2

2N + 1

]

,

C1 =b3h12

Es

[

r2 +2r(1− r)

N + 1+

(1− r)2

2N + 1

]

,

C2 =bh3

12Es

[

r2 +6r(1− r)

N + 3+

3(1− r)2

2N + 3

]

,

A1 = κbh Gsr + 3

N+3 (1−r)

r +(

1N+1 + JN

)

(1−r)

[

r2 +2r(1−r)

N+1+

(1−r)2

2N+1

]

,

A2 = κbh Gs

[

r2 +2r(1− r)

N + 1+

(1− r)2

2N + 1

]

.

The values have been verified against numerical solutions for:• cantilever beams with uniformly distributed force;• cantilever beams with end concentrated force;• three-point bending of FGM beam.


Three-point bending of a FGM beam

l

P

@@@

δ

The analytical solution for the maximum deflection is

δexact=P l4

( 1A1

+l2

12C2

)

.

We solve the problem numerically by the finite element method.Consider the concentrated force P = 5 kN at the mid-span.

Dimensions: length l = 1 m ; thickness h = 5 cm, b = 5 cm.Material: ρm = 500 kg

m3 , ρs = 2700 kgm3 , Es = 70 GPa, ν = 0.3;

with the exponent: N ∈ {1, 2, ..., 10}.


Comparison between analytical solutions and numerical results

fig 5: Displacement for representative values exponent N. (three point bending beam)

Theoretical models has got good results for cantilever beam.If static diagram is more complicated, we should use to more layers in FEA model.

-7-

0 0.2 0.4 0.6 0.8 1 1.2

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

x3 [m]

[mm]Deflection

N = 1

N = 5

N = 10

The relative error is in the range : 0.9 - 2.9 % , for N = 1, 2, ..., 10.


Sandwich beams with foam core

btf

c

tf

O

x2

x1

ad

The stiffness properties which are of special interest forsandwich beams are:• the bending effective stiffness C1 about the Ox1 axis ;• the shear effective stiffness A2 in the x2 direction ;• the torsional rigidity C3 .


Bending stiffness and shear stiffness

• Effective bending stiffness :

C1 = Efb tf d2

2+ Ef

b t3f6

+ Ecb c3

12,

which coincides with the classical results from Allen(1969),Zenkert(1997), Gibson & Ashby (1997).

• Effective shear stiffness :

A2 = κ12ba2

c Gc + 2tf Gf

cρc + 2tf ρf + c(ρf −ρc)F(π c2a )

(

ρftf d2

2+ ρf

t3f6+ ρc

c3

12

)

where the function F( · ) is defined on the interval (−π2,π

2) by :


F(x) =cosx

xln(1+ sinx

cosx

)

for x 6= 0, F(0) = 1.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

In the case of thin faces , we get the approximate value

A2 = κb(

c Gc + 2tf Gf + 4tf Gcρf − ρc

ρc

)

.

and for very thin faces ( tf ≪ c ) we obtain the approximation

A2 = κb c Gc ,

which corresponds to the classical value (Allen 1969).


Comparison with experimental results• Three-point bending of sandwich beams

Figure: a) MTS 25 kN testing machine; b) specimens of sandwichbeams used in experiments

We compare the maximum deflection D for theoreticalpredictions and experimental measurements.


Two types of sandwich beams with polyurethane foam core:• beams with polyester faces;• beams with epoxy faces.

For foams with closed cells (Gibson & Ashby 1997)

Ec

Es= φ2

(ρc

ρs

)2+ (1− φ)

ρc

ρs,

Gc

Es=

38

[

φ2(ρc

ρs

)2+ (1− φ)

ρc

ρs

]

,

where φ is the volume fraction of solid material contained inthe cell edges. For rigid polyurethane foams we have φ = 0.8.

Face Core tf c Ef Ec Gf Gc ρf ρc

[mm] [mm] [MPa] [MPa] [MPa] [MPa] [ Kgm3 ] [ Kg

m3 ]

Polyester Foam 0.1 12 4000 74.8 1400 28.05 1200 200

Epoxy Foam 0.17 12.03 3400 74.8 1353.5 28.05 1060 200


0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5

D [mm]

P [N]

Experiments

Approximate solution

Classical solution

@@Exact analytical solution

Figure: Beam with epoxy faces and polyurethane foam core


0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5 3

D [mm]

P [N] @@@

Experiments

Approximate solution @@

Exact analytical solution

Classical solution

Figure: Beam with polyester faces and polyurethane foam core


Multilayered beams

O x1

Si

S2

S1

Sn

zizi−1

x2

z0

zn

Consider multilayered beams with n layers, having• different thicknesses ti• different material properties Ei , Gi , ...


Torsional rigidity

C3 =b3

3

(

n∑

i=1

Gi ti)

+ 4n

∑

i=0

[

(Gi+1−Gi)

∞∑

k=0

(−1)k

m2

(

A(i)k sinh(mzi)+B(i)

k cosh(mzi))

]

where m= (2k+1)πb , and A(i)

k , B(i)k are given by the system:

A(j)k sinh(mzj) + B(j)

k cosh(mzj) = A(j+1)k sinh(mzj) + B(j+1)

k cosh(mzj),

Gi[

A(i)k cosh(mzi)+B(i)

k sinh(mzi)]

= Gi+1[

A(i+1)k cosh(mzi)+B(i+1)

k sinh(mzi)]

+ (Gi+1 − Gi)(−1)k 8b2

π3(2k+1)3 .

In the case of 2 or 3 layers ( n = 2, 3 ), the formula for the torsionalrigidity C3 reduces to known results from the literature.


Flexural rigidity

From the relation (EI)eq = C1 −B2

31

A3we find

the equivalent flexural rigidity :

(EI)eq = b[

n∑

k=1

t3kEk

12+

(

n∑

k=1

tkEk

)−1 ∑

1≤k<l≤n

tkEk · tlEl · (mk − ml)2]

where the difference mk − ml =tk2+ tk+1 + ...+ tl−1 +

tl2

is the distance between the middle planes of the layers.

This is a generalization of known results for the case n = 3.


Sandwich columns

l2R2 2R1 O x1

x2

(ρc ,Ec , νc)

(ρf ,Ef , νf )

We consider 2 types of circular sandwich columns:• piecewise homogeneous ;• functionally graded with exponential distribution law.


Functionally graded sandwich columns

The core has exponential distribution law in radial direction

ρc = ρ0 exp(−σ r), λc = λ0 exp(−σ r), µc = µ0 exp(−σ r).

The face is made of another functionally graded material

ρf = ρ(r), Ef = E(r), νf = ν0 (constant).

• Effective extensional stiffness and bending stiffness :

A3 = 2π∫ R2

R1

rE(r)dr +2πσ2 E0

[

1− (1+ σR1)e−σR1

]

,

C1 = C2 = π

∫ R2

R1

r3E(r)dr+π

σ4 E0[

6−(6+6σR1+3σ2R21+σ

3R31)e

−σR1]

.


• Torsional rigidity :

C3 =π

1+ ν0

∫ R2

R1

r3E(r)dr+2πσ4 µ0

[

6−(6+6σR1+3σ2R21+σ

3R31)e

−σR1]

• Effective shear stiffness :

A1 = A2 =2π κ

R22

·

11+ ν0

∫ R2

R1

rE(r)dr +2σ2 µ0

[

1− (1+ σR1)e−σR1

]

∫ R2

R1

r ρ(r)dr +1σ2 ρ0

[

1− (1+ σR1)e−σR1

]

×

×{

∫ R2

R1

r3ρ(r)dr +1σ4 ρ0

[

6− (6+ 6σR1 + 3σ2R21 + σ3R3

1)e−σR1

]

}


Comparison with numerical resultsNumerical example: length l = 20 cm ; radii R1 = 1.25 , R2 = 1.5 cm.The core: FGM alumina foam E0 = 343GPa, ν0 = 0.3, ρ0 = 3880kg/m3; with the exponent: σ ∈ {25, 50, 75, 100, 150}.The face (skin): homogeneous (aluminum alloy) Ef = 70 GPa andρf = 2688kg/m3.

0 0.0025 0.005 0.0075 0.01 0.0125 0.0150

50

100

150

200

250

300

350

E

r [m]

[GPa]

σ = 25

σ = 50

σ = 75

σ = 100

σ = 150

Figure: The distribution of Young modulus E in the core.


Bending of cantilever FGM beam by end loads

(a) (b)

l

P

u

F

lx1

x3O

δ

The concentrated end force P = 616N .The analytical solution for the maximum deflection is

δexact= P l( 1

A1+

l2

3C2

)

,


Finite element analysis with Abaqus :

Comparison of results :

σ 25 50 75 100 150

δexact [mm] 0.2527 0.3045 0.3618 0.4237 0.5551δFEM [mm] 0.2523 0.3034 0.3600 0.4209 0.5502

Error ∆ [%] -0.1719 -0.3532 -0.5012 -0.6658 -0.8962


Extension of FGM sandwich column

The theoretical axial displacement : uexact=F lA3

.

Finite element analysis with Abaqus :

fig 7: Odkształcenia górnej powierzchni modelu przy rozci ganiu


Comparison of results :

σ 25 50 75 100 150

uexact [mm] 0.0929 0.1110 0.1314 0.1542 0.2059uFEM [mm] 0.09354 0.1117 0.1321 0.1547 0.2060

Error ∆ [%] 0.7073 0.587 0.4654 0.3417 0.0846

Conclusion :

The good agreement between the analytical solutions and thenumerical results and classical previously known results showsthat the formulas for effective stiffness coefficients are correct.


Shells and Plates


Kinematical model of directed surface(simple shells)

O

(x1, x2) (x1, x2)

S0

S

rR

d1

d2d3 D1

D2D3

• Each point is connected to a triad of rigidly rotating directors

• The reference configuration S0 : { r(x1, x2) ; dk(x1, x2) }• The actual configuration S : {R(x1, x2, t) ; Dk(x1, x2, t) }


• The rotation tensor P and angular velocity vector ω are

P(x, t) = Dk(x, t) ⊗ dk(x), ω = axial(

P · PT) .

• We define the vector : Φα(x, t) = axial(

∂αP · PT)

.

In the linear theory, we introduce :

• the displacement vector u :

u(x, t) = R(x, t) − r(x)

• the vector of infinitesimal rotations ϕ :

ϕ(x, t) = ω(x, t), ∂αϕ(x, t) = Φα(x, t).


Strain tensors

• the fundamental tensors of the surface S0 :

a ≡ ∇r, b ≡ −∇n, c ≡ −a × n.

The infinitesimal strain tensors are :

ε =12

(

e · a + a · eT)

, e = ∇u + a ×ϕ,

γ = e · n , k = (∇ϕ) · a +12(e · ·c)b,

• the symmetric tensor ε : extensional strains and in–planeshear strains

• γ accounts transverse shear deformation

• k is a tensor of bending and twist strains.


Equations of motion

The equations of motion in local form :

∇ · T + F = ρ(

u +ΘT1 · ϕ

)

,

∇ · M + T× + L = ρ (Θ1 · u +Θ2 · ϕ) .

whereT is the force tensor, M is the moment tensor,ρ is the reference mass density,F, L are the external body loads, andρΘα represent tensors of inertia.

Thermal effects : denote by

T1(x, t) and T2(x, t)

the temperature fields on the two sides of the shell at time t.


Temperature distribution

The variation of temperature across the thickness is T1 − T2

and the average temperature through the thickness is T1+T22 .

The temperature variations τ1 , τ2 are given by:

τ1(x, t) = T1(x, t)− T0 , τ2(x, t) = T2(x, t)− T0 .

T1(x1,x2,t)

T2(x1,x2,t)


The equations of heat transfer :

∇ · hγ + ρT0Sγ = ρ(gγ + Qγ + g0γ), γ = 1,2,

where

• Sγ is the entropy density function associated to side γ

• gγ is the heat exchanged with the surrounding medium

• g0 = g01 + g02 is the heat supply inside the surface S0

• Q ≡ Q1 = −Q2 represents the heat exchanged between sides

• hγ are the heat flux vectors :

h1 = −γ1 · ∇τ1 , h2 = −γ2 · ∇τ2 ,

where γ1 , γ2 are the thermal conductivity tensors.


The constitutive equations of thermoelastic shells fororthotropic and inhomogeneous materials:

Ψ(ε,γ, k, τ1, τ2) = U(ε,γ, k) + ΨT(ε, k, τ1, τ2) ,

ρU =12ε · ·C1 · ·ε+ ε · ·C2 · ·k +

12

k · ·C3 · ·k +12γ · Γ · γ ,

ρΨT = ε · ·(

C4 τ1 + C5 τ2)

+ k · ·(

C6 τ1 + C7 τ2)

− 12α1τ

21 − 1

2α2τ

22 .

• The tensors C1 , C2 , C3 and Γ represent the stiffness tensorswhich characterize the effective elastic properties of the shell.• The tensors Ck (k = 4, ...,7) and the coefficients α1, α2

express the coupling between thermal and elastic fields.

S =∂(ρΨ)

∂ε, N =

∂(ρΨ)

∂γ, M =

∂(ρΨ)

∂k, Sγ = − ∂Ψ

∂τγ(γ = 1,2).


Concerning the associated boundary–initial–value problem,we have proved:

Uniqueness of solution ; theorem of power and energy

Reciprocity relations

Variational theorems

Continuous dependence of solutions upon initial data

Continuous dependence on the external loads andheat supply

Inequalities of Korn type for directed surfaces

Existence results for the equilibrium equations

Existence of weak solutions to the dynamical equations


Variational TheoremWith the help of the convolution product and the functionse(t) = 1, j(t) = t, we can incorporate the initial conditions intothe field equations in the following way

j ∗(

∇·T)

+ ρF = ρ(

u +ΘT1 ·ϕ

)

,

j ∗(

∇·M + T×

)

+ ρL = ρ(

Θ1·u +Θ2·ϕ)

,

e∗[

∇ · hγ − ρ(gγ + Qγ)]

− ρVγ = −ρT0Sγ , γ = 1,2,

where we denote

F = j ∗ F +(

u0 + tv0)

+ΘT1 ·

(

ϕ0 + tψ0

)

,

L = j ∗ L +Θ1 ·(

u0 + tv0)

+Θ2 ·(

ϕ0 + tψ0

)

,

Vγ = e∗ g0γ + T0S0γ .


Theorem

For any t ∈ T , we define the functional Λt(·) by

Λt(s) =12

∫

S0

{

j∗[

S∗ε+ MT∗ k + N∗γ − ρSγ ∗τγ +eT0

∗(

hγ∗∇τγ

+ρ(gγ + Qγ + βγ τγ) ∗ τγ)]

+ ρ[

u ∗ u + 2ϕ ∗ (Θ1 · u)

+ϕ ∗ (Θ2 · ϕ)− 2(

F ∗ u +L ∗ϕ− 1T0

j ∗ Vγ ∗ τγ)]

}

da

−∫

C2

j ∗ (t ∗ u + m ∗ϕ)dl − 1T0

∫

C4

e∗ j ∗ hγ ∗ τγ dl,

for any s={

u,ϕ, τγ , ε,γ, k,T,M,Sγ ,hγ ,gγ ,Qγ

}

kinematicallyand thermally admissible process . Then, we have

δΛt(s) = 0, ∀ t ∈ T

if and only if the field v = {u,ϕ, τγ} is a solution of the problem.


Existence of weak solutions

We apply the method of semigroup of operators to thedynamical equations.Define the Hilbert space:

Z ={

U = (ui , vi , ϕβ , ψβ, τγ) ; ui , ϕβ ∈ H10(Σ), vi , ψβ , τγ ∈ L2(Σ)

}

,

with the scalar product

⟨

U,V⟩

Z =12

∫

Σρ(

v · v +ψ ·Θ1 · v + ψ ·Θ1 · v + ψ ·Θ2 · ψ)√

adx1dx2

+12

∫

Σ

[

T(z)··ε(y) + MT(z)··k(y) + N(z)·γ(y) + ρSγ(y) τγ]√

adx1dx2,

where U = (ui , vi , ϕβ , ψβ, τγ), V = (ui , vi , ϕβ , ψβ , τγ),y = (ui , ϕβ , τγ), z= (ui , ϕβ , τγ).


Define the operator A : D(A) ⊂ Z → Z given by its components

AU = (vi , BiU , ψα , CαU , HγU) , ∀U = (ui , vi , ϕα, ψα, τγ),

BαU =ρ

ρρ2 − (ρ1)2 rα ·[

Θ2 · (∇ · T) +Θ1 · (∇ · M + T×)]

(ui , ϕβ , τγ),

B3U =1ρ

n ·[

∇ · T]

(ui , ϕβ , τγ),

CαU =ρ

ρρ2 − (ρ1)2 rα ·[

ΘT1 · (∇ · T) + (∇ · M + T×)

]

(ui , ϕβ , τγ),

H1U =1α1

{(

C4··ε+ C6··k)

(vi , ψβ) +1T0

[

∇·(γ1·∇τ1)−ρβ1τ1 −ρβ(τ1−τ2)]}

,

H2U =1α2

{(

C5··ε+ C7··k)

(vi , ψβ) +1T0

[

∇·(γ2·∇τ2)−ρβ2τ2 −ρβ(τ2−τ1)]}

.

The domain D(A) of the operator is the set

D(A) ={

U ∈ Z ; AU ∈ Z , τγ ∈ H10(Σ)

}

.


We can write our boundary–initial–value problem as aCauchy problem: find U = (ui , vi , ϕβ , ψβ , τγ) ∈ D(A) such that

ddt

U(t) = AU(t) + F(t), with U(0) = U0 , (1)

where U0 – initial data, and F(t) – external loads / heat supply.

Lemma

The operator A : D(A) ⊂ Z → Z has the properties:(i) The domain D(A) of the operator is dense in Z;(ii) The operator A satisfies

〈AU,U〉Z ≤ 0, ∀U ∈ D(A) ;

(iii) A verifies the range condition

Range(I − A) = Z .


We can apply the Lumer–Phillips theorem and obtain thefollowing existence result for the weak solution.

TheoremAssume that the external body loads and heat supply are suchthat F(t) ∈ C1([0, t0],L2(Σ)) and the initial data satisfyU0 ∈ D(A).Then, there exists a unique solutionU(t) ∈ C1([0, t0],Z) ∩ C0([0, t0],D(A)) for the problem (1).The unique solution U(t) satisfies the estimate

‖U(t)‖Z ≤ ‖U0‖Z +

∫ t

0‖F(s)‖Z ds, t ∈ [0, t0].

Remark. This estimate shows the continuous dependence ofthe weak solution U(t) with respect to the initial data U0 and theexternal body loads and heat supply F(t).


Applications : Layered Thermoelastic Shells

The Helmholtz energy function Ψ is given by:

Ψ(ε,γ, k,T1,T2) = U(ε,γ, k) + ΨT(ε, k,T1,T2),

ρU = 12 ε··C1··ε+ ε··C2··k + 1

2 k··C3··k + 12 γ · Γ · γ,

ρΨT = −ε··(C4T1+C5T2) + k··(C6T1+C7T2)− 12 α1T2

1− 12 α2T2

2 .

The effective stiffness tensors :

• C1 ,C2 ,C3 ,Γ characterize the elastic properties (known);• C4 ,C5 ,C6 ,C7 are the thermo-elastic properties (new).

We denote by rα = ∂αr and the tensor basis :

a1 = r1 ⊗ r1 + r2 ⊗ r2 , a2 = r1 ⊗ r1 − r2 ⊗ r2 ,a3 = r1 ⊗ r2 − r2 ⊗ r1 , a4 = r1 ⊗ r2 + r2 ⊗ r1 .


The structure of the thermo-elastic tensors C4 ,C5 ,C6 ,C7 is:• for symmetrical multi-layered shells (transversal isotropy):

C4 = C5 = C4 a1 , C6 = −C7 = C6 a3 ;

• for non-symmetrical layered shells (transversal isotropy):

C4 = C4 a1 , C5 = C5 a1 , C6 = C6 a3 , C7 = C7 a3 .

Our aim is to determine the ‘thermal’ coefficients C4 , C5 , C6 , C7 .

Method of determination:

analysis of deformation of cylindrical multi-layered shells;general analytical solution procedure

comparison with 3D solutions for thermal stresses problems andidentification of constitutive coupling coefficients

validation of results by comparison with numerical solutions forthermoelastic shells


Homogeneous and isotropic shellsThe elastic effective stiffness coefficients are (Zhilin 2006):

A11 =E h

1− ν, A22 = A44 = 2G h, C33 =

E h3

3(1− ν),

C22 = C44 =E h3

3(1+ ν), Γ1 = 2G hΓ0 (Γ0 = π2

12),

where the thickness of the shell is 2h .For thermo-elastic coefficients we have obtained :

C4 = αE h

1− ν= β h

1− 2ν1− ν

C6 = αE h2

3(1− ν)= β

h2

31− 2ν1− ν

α= coefficient of thermal expansion ; β= stress-temperat. modulus.


Numerical example: Cylindrical homogeneous shells

Material of the shell : single crystal Ni–based superalloyE = 15 GPa, ν = 0.333, α = 16.8 · 10−6 K−1

Temperature distribution :

T1(s) = T2(s) =T0

2sin

sr0


Deformation of the shell (FEM calculations with ABAQUS):


Pt. z Result uaxial uradial utang ∆axial ∆radial ∆tang

[m] type [mm] [mm] [mm] [%00] [%00] [%00]

0 Theor. 0.000 -0.039 0.177 0.000 26.28 1.774FEM 0.000 -0.040 0.177

0.25 Theor. 0.148 -0.132 0.085 0.061 3.352 0.846FEM 0.149 -0.132 0.085

0.5 Theor. 0.297 -0.410 -0.194 0.064 1.038 1.938FEM 0.297 -0.411 -0.194

0.75 Theor. 0.445 -0.874 -0.658 0.159 0.448 6.578FEM 0.445 -0.875 -0.658

1 Theor. 0.594 -1.524 -1.307 0.104 0.157 13.07FEM 0.594 -1.524 -1.307

Table: Comparison of displacements obtained in the theoretical andnumerical approaches, in five 5 different points of the shell.


Three-layered shells (symmetrical)

h22h1

Effective elastic moduli for 3-layered shells (Zhilin, 2006)

A11 =E1h11−ν1

+ E2h21−ν2

, A22 = A44 =E1h11+ν1

+ E2h21+ν2

, A12 = 0,

C33 =E1h3

13(1−ν1)

+E2(h3−h3

1)3(1−ν2)

, C22 = C44 =E1h3

13(1+ν1)

+E2(h3−h3

1)3(1+ν2)

,

where the thickness of the shell is 2h = 2(h1 + h2).


For thermo-elastic coefficients we have obtained :

C4 = α1E1 h1

1− ν1+ α2

E2 h2

1− ν2

C6 =13h

(α1E1 h31

1− ν1+α2E2(h3 − h3

1)

1− ν2

)

Alternative formulas:

C4 = β1 h11− 2ν1

1− ν1+ β2 h2

1− 2ν2

1− ν2,

C6 =13h

[

β1 h31

1− 2ν1

1− ν1+ β2(h

3 − h31)

1− 2ν2

1− ν2

]

,

where α1 , β1 , E1 , ν1 , h1 are parameters for the interior layer,and α2 , β2 , E2 , ν2 , h2 are parameters for the exterior layer.


Application: Shell with thermal barrier coating (TBC)

Material of the shell :

The TBC layers are made of yttria stabilized zirconia (YSZ)with typical composition ZrO2–6-8 wt% Y2O3 andthermoelastic coefficients (measured at 1000◦C)

E2 = 35 GPa, ν2 = 0.1, α2 = 10 · 10−6 K−1 .

The interior layer : single crystal Ni-based superalloy


Distribution of the temperature:

Case 1: τ1 = 150, τ2 = 1200z+ 150;

Case 2: τ1 = 200z+ 200, τ2 = 1000z+ 200;

Case 3: τ1 = 100z2 + 150z+ 200, τ2 = 500z2 + 300z+ 200.

Thermal Axial displ. [mm] ∆ Radial displ. [mm] ∆load Theor. FEM [%00] Theor. FEM [%00]

Case 1 2.1705 2.1706 0.046 1.3023 1.3023 0.000Case 2 2.5323 2.5324 0.039 1.4470 1.4470 0.000Case 3 2.3624 2.3623 0.042 1.3021 1.3022 0.076

Table: Comparison of axial and radial displacements in the positionz= z/2, obtained in the analytical and numerical approaches.


0 0,2 0,4 0,6 0,8 1 1,2

000,0E-2

500,0E-6

100,0E-5

150,0E-5

200,0E-5

250,0E-5

300,0E-5

350,0E-5

Z [m]

U.r

ad

ial

[m]

0 0,2 0,4 0,6 0,8 1 1,2

000,0E-2

100,0E-5

200,0E-5

300,0E-5

400,0E-5

500,0E-5

600,0E-5

700,0E-5

800,0E-5

FEATHEOR

Z [m]

U.a

xial

[m]

(a) (b)

Figure: Comparison of theoretical results (THEOR) and finite elementsolutions (FEA) for the 3-layered cylindrical shell:a) axial displacements; b) radial displacements.


Two-layered plates (non-symmetrical)We denote the thicknesses of the two layers by h1 and h2 , andthe material parameters by α1 , β1 , E1 , ν1 and α2 , β2 ,E2 , ν2 .We find the 4 different thermo-elastic coefficients :

C4 =α2 E2 h2

1− ν2+

12h

(α1 E1 h21

1− ν1− α2 E2 h2

2

1− ν2

)

C5 =α1 E1 h1

1− ν1− 1

2h

(α1 E1 h21

1− ν1− α2 E2 h2

2

1− ν2

)

C6 =α2 E2 h2

2

3(1− ν2)− h1

6h

(α1 E1 h21

1− ν1− α2 E2 h2

2

1− ν2

)

C7 = − α1 E1 h21

3(1− ν1)− h2

6h

(α1 E1 h21

1− ν1− α2 E2 h2

2

1− ν2

)

where h = h1 + h2 is the thickness of the plate.


Comparison with numerical solutions:Rectangular plate

Consider a rectangular plate with 2 layers:

carbon steel 1%, with thickness h1 = 2 mm and materialparameters

E1 = 205 GPa, ν1 = 0.3, α1 = 1.3 · 10−5 K−1

wrought aluminum alloy 1060 of thickness h2 = 0.5 mm

E2 = 69 GPa, ν2 = 0.33, α2 = 2.35 · 10−5 K−1

Bending due to the difference of temperature on the surfaces:

T1 = −10◦C, T2 = 60◦C.


Numerical solution with ABAQUS:2

Figure: The deformed shell and the distribution of transversaldisplacements (U3).


Coord. [mm] u1 [mm] u2 [mm] u3 [mm]x2 x3 Theor. FEM Theor. FEM Theor. FEM

47.5 0 0.002 0.002 0 0 0.419 0.41923.75 100 0.001 0.001 .0042 .0042 1.965 1.965

0 150 0 0 .0063 .0063 4.186 4.18647.5 210 0.002 0.002 .0088 .0088 8.625 8.625

Table: Comparison of displacements for 4 the points of the plate, inthe two approaches: analytical (Theor.) and numerical (FEM).

The theoretical and numerical results coincide.

Conclusions :

The theory is mathematically well–formulated

The formulas for coupling coefficients are appropriate

They can be used to treat thermoelastic shell problems.


Nonlinear 6–Parameter Shells


Kinematical model of 6–parameter resultant shells

The reference (initial) configuration of the shell is given bythe position vector y0 and the structure tensor Q0 :

y0 : ω ⊂ R2 → R

3, y0 = y0(x1, x2),

Q0 : ω ⊂ R2 → SO(3), Q0 = d0

i (x1, x2)⊗ ei ,

The orthonormal triad of initial directors : {d01, d

02, d

03}

describes the orthogonal structure tensor Q0.

The deformed configuration is characterized by

y = χ(y0), Qe = di ⊗ d0i ∈ SO(3),

where χ is deformation of the base surface and theorthogonal tensor Qe is the (effective) elastic rotation.

y denotes the position vector and {d1, d2, d3} theorthonormal triad of directors in the deformed configuration


Figure : The base surface of the shell

e1

e2

x1

x2

e3

ω

y0(x1, x2) R(x1, x2)

y(x1, x2)

Q0(x1, x2)

d01

d02

d03

d1

d2d3χ(y0)

Qe(y0)


Equilibrium equations and boundary conditions

Grads and Divs are the surface gradient andsurface divergence operators and F = Gradsy = ∂αy⊗ aα

denote the shell deformation gradient tensor.

Equations of equilibrium for 6-parameter shells :

Divs N + f = 0, Divs M + axl(NFT − FNT) + c = 0,

where N and M are the internal surface stress resultantand stress couple tensors, while f and c are the externalsurface resultant force and couple vectors.

We consider boundary conditions of the type

Nν = n∗, Mν = m∗ along ∂S0f ,

y = y∗, R = R∗ along ∂S0d .


Nonlinear elastic strain and curvature measuresThe elastic shell strain tensor Ee in material representation

Ee = Qe,TGrads y − Grads y0.

It is useful in the proof to express the elastic shell straintensor in terms of the total rotation R and initial rotation Q0 :

Ee = Q0(RT∂αy − Q0,T∂αy0)⊗ aα .

The elastic shell bending-curvature tensor Ke :

Ke =[

Qe,Taxl(∂αRRT)− axl(∂αQ0Q0,T)]

⊗ aα .

or, in terms of the total and initial rotations R,Q0 :

Ke = K − K0, K = Q0axl(RT∂αR)⊗ aα,

K0 = axl(∂αQ0 Q0,T)⊗ aα,

where K is the total bending–curvature tensor, while K0 isthe given initial bending-curvature tensor.


Variational formulation for elastic shells

Let W = W(Ee,Ke) denote the strain energy density.

The constitutive equations (hyperelasticity assumption):

N = Qe ∂W∂Ee , M = Qe ∂W

∂Ke .

where N is the internal surface stress resultant andM is the stress couple tensor.

W is assumed to be a quadratic function of Ee and Ke :the model is physically linear and geometrically non-linear.

Two–field minimization problem: find the pair (y, R) in theadmissible set A which realizes minimum of the functional

I(y,R) =∫

S0W(Ee,Ke)dS− Λ(y,R) for (y,R) ∈ A.


The admissible set A is defined by

A ={

(y,R) ∈ H1(ω,R3)×H1(ω,SO(3))∣

∣ y∣∣∂S0

d

= y∗, R∣∣∂S0

d

= R∗}

.

The function Λ(y,R) represents the potential of externalsurface loads f , c, and boundary loads n∗, m∗

Assume that the external loads satisfy the conditions

f ∈ L2(ω,R3), n∗ ∈ L2(∂ωf ,R3),

and the boundary data satisfy the regularity conditions

y∗ ∈ H1(ω,R3), R∗ ∈ H1(ω,SO(3)).


Main result: Existence of minimizers

Theorem

Assume that the initial configuration satisfies: y0 : ω ⊂ R2 → R

3

is a continuous injective mapping and

y0 ∈ H1(ω,R3), Q0 ∈ H1(ω,SO(3)),

∂αy0 ∈ L∞(ω,R3), det(

aαβ(x1, x2))

≥ a20 > 0 .

The strain energy density W(Ee,Ke) is assumed to be aquadratic, convex and coercive function of (Ee,Ke):

W(Ee,Ke) ≥ C0(

‖Ee‖2 + ‖Ke‖2 ).

Then, the minimization problem admits at least one minimizingsolution pair (y, R) ∈ A.


Sketch of the Proof

We employ the direct methods of the calculus of variations.

We show first that there exists C > 0 such that

|Λ(y,R) | ≤ C(

‖y‖H1(ω) + 1)

, ∀ (y,R) ∈ A.

Using this relation and the coercivity of W we obtain

I(y,R) ≥ C0 ‖∇y‖2L2(ω) − C1‖ y ‖H1(ω) − C2 .

Applying the Poincaré–inequality we infer

I(y,R) ≥ cp ‖y−y∗‖2H1(ω)−C3‖y−y∗‖H1(ω)+C4 , ∀ (y,R) ∈ A

so that the functional I(y,R) is bounded from below over A.


There exists an infimizing sequence (yn,Rn) such that

limn→∞

I(yn,Rn) = inf{

I(y,R)∣

∣ (y,R) ∈ A}

.

The sequences{

yn

}

and{

Rn}

are bounded in H1(ω).

We can extract subsequences (not relabeled) such that

yn ⇀ y in H1(ω,R3) and yn → y in L2(ω,R3),

Rn ⇀ R in H1(ω,R3×3) and Rn → R in L2(ω,R3×3).

Next we show the weak convergence (on subsequences)

Een ⇀ Ee in L2(ω,R3×3) and Ke

n ⇀ Ke in L2(ω,R3×3).


Finally, we use the convexity of the strain energy W :∫

ω

W(Ee, Ke)adx1dx2 ≤ lim infn→∞

∫

ω

W(Een,K

en)adx1dx2,

which implies weak lower semi-continuity :

I(y, R) ≤ lim infn→∞

I(yn,Rn).

Since (yn,Rn) is an infimizing sequence, we obtain:

I(y, R) = inf{

I(y,R)∣

∣ (y,R) ∈ A}

,

i.e., (y, R) is a minimizing solution pair.

Remark. The existence theorem is applicable in the cases:

Isotropic shells

Orthotropic shells

Composite layered shells


Future Plans


We intend to investigate the following topics, which are ofgreat importance in the context of shell theory:

plastic or viscoplastic materials

damage and fracture mechanics in shells

shells made from materials with memory: viscoelasticshells and rods

elastic solids with surface stress (prospectiveapplication of the shell theory)

theory of mixture applied to composite thinmechanical structures


Thank you for your attention !

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Research Activities and Scientiﬁc Results of Mircea BÎRSAN · 2015-08-19 · Research Activities...

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