J. N. Reddy Center for Innovations in Mechanics for Design and - - PowerPoint PPT Presentation

JN Reddy

• J. N. Reddy

Center for Innovations in Mechanics for Design and Manufacturing Texas A&M University, College Station, Texas jnreddy@tamu.edu; http://mechanics.tamu.edu City U Distinguished Lecture

City University of Hong Kong 12 October, 2018

JN Reddy

• Professional retrospectives:
• Primal-dual variational principles (PhD)
• Hypervelocity impact (Lockheed)
• Modeling of geological phenomena (OU)
• Modeling of bimodular materials (OU)
• Third-order structural theories (VPI)
• Penalty finite element models of flows of viscous

incompressible fluids (VPI)

• Layerwise laminate theory (VPI & TAMU)
• Robust shell element (TAMU)
• Modeling of biological cells (TAMU)
• Least-squares FE models of fluid flow (TAMU)
• Strain gradient, non-local, and non-

classical continuum models (TAMU)

• GraFEA (TAMU)
• Closing remarks

5

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Professor J. Tinsley Oden, Director, Institute for Computational Engineering and Sciences; Professor of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin (JN’s Ph.D. Thesis advisor and coauthor of papers and books).

3rd ed. to appear in 2017

JN Reddy

J.T. Oden and J.N. Reddy, “On dual-com plem entary variational principles in m athem atical physics,” Int. J. Engng Science, 12, 1-29 (1974). Supported by AFOSR

• Variational principles for
• 14 Variational principles of elasticity: 7 primal and 7 dual;
• Fluid mechanics, electrostatics, magnetostatics; and

nonlinear operators

*

( ) 0 in and T ET u f + = Ω

*

( ) 0 in S CS c h + = Ω

All conventional as well as m ixed variational principles are

• derived. Several of these principles form ed the basis of the

m ixed, hybrid, and assum ed strain finite elem ent m odels (they were not cited often because our work was a bit m athem atical and buried in the literature).

8

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( , , )

xz x y z

s

z

55 45

( , , ) ;

xz xz yz xz yz

x y z Q Q u w v w z x z y s g g g g = + ∂ ∂ ∂ ∂ = + = + ∂ ∂ ∂ ∂

Transverse shear stress

2 3 2 3

( , , ) ( , , ( , ) ( , ) ( ) , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ) , , (

y x y y x x

x y x y x y x y x u x y z u z z z v x y z v z z z y x w y x y x x y z w y x y f q l f q l = + + + = + + + =

Displacem ent field

J.N. Reddy, “A sim ple higher-order theory for lam inated com posite plates,” J. of Ap p lied Mecha nics, 51, 745-752 (198 4). (over 20 0 0 citations) J.N. Reddy and C.F. Liu, “A higher-order shear deform ation theory for lam inated elastic shells,” Int.

• J. of Engng. Sci., 23(3), 319-330 (198 5). (8 0 0 citations)

CLPT FSDT TSDT

12

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xz

s z

2 2

( , ) 3 ( , ) ( , ) 3 ( , ) ( ( ) , ) , 2 2

x xz y y z x y x y

w x y z x y x w x y z u w x y z x v w x y z z z y y x y g g q f l l q f + + ∂ ∂ = + = ∂ ∂ ∂ ∂ = + = ∂ + ∂ ∂ + ∂ ∂ + ∂ +

Transverse shear strains Vanishing of transverse shear stresses on the bounding planes

2 2

( , , / 2) ( , , / 2) ( , ) ( , ) 4 4 ( , ) ( , ) , ( , ) ( , ) 3 3

y x x y y xz yz x

x y h x y h x y x y w w x y x y x y x y h x h y l s s f f q l q ± = ± =  = = ∂ ∂     = − + = − +     ∂ ∂    

3 2 3 2

4 3 ( , , ) ( , , ) ( ( , ) ( , ) ( , ) ( , ) , 3 ) , ( , 4 )

x x y y

u x y z u z v x y z v z w x y z w z x y x y x y w h x z w x y x y h y f f f f = + = ∂   − +   + = ∂   ∂   − +   ∂  

∂x ∂w φx (u,w) u0 w ( , )

TSDT

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5 10 15 20 25 30 35 40 45 50

a/h

0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020

Deflection, w

_

TSDT 3-D Elasticity Solution FSDT CLPT

Bending of a symmetric cross-ply (0/90)s laminate (SS-1)under uniformly distributed load

FSDT TSDT

Third-order laminate theory: 13

E1=25E2 , G12=G13=0.5E2 G23=0.2E2 , n12=0.25 E2 = 10 6 psi (7 GPa)

3D CLPT

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0.00 0.04 0.08 0.12 0.16 0.20

Stress, σ

yz (a/2,0,z)

• 0.50
• 0.30
• 0.10

0.10 0.30 0.50

CLPT (E) FSDT (E) FSDT (C) TSDT (C) TSDT (E)

(E): equilibrium-derived (C): constitutively-derived

Bending of a symmetric cross-ply (0/90)s laminate (SS-1)under uniformly distributed load CLPT FSDT (C) TSDT TSDT (C) FSDT (E)

Third-Order Laminate Theory 16

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Equilibrium of interlaminar stresses

σzx

(k)

σzy

(k)

σzz

(k)

k

kth layer (k+1)th layer

= σzx

(k+1)

σzx

(k)

= σzy

(k+1)

σzy

(k)

= σzz

(k+1)

σzz

(k)

k+1

σzx

(k+1)

σzy

(k+1)

σzz

(k+1)

y x z 9 1 k k xx xx yy yy xy xy

s s s s s s

+

        ≠            

JN Reddy

Single-Layer Theories Equilibrium Requirements

10 1 1 k k k k zz zz zz zz yz yz yz yz xz xz xz xz

s s e e s s e e s s e e

+ +

                =  ≠                        

( ) ( 1)

because

k k ij ij

Q Q

+

≠

kth layer (k+1)th layer

y x z

1 1

,

k k k k xx xx zz zz yy yy yz yz xy xy xz xz

e e e e e e e e e e e e

+ +

                = =                        

u(x, y, z, t) =

N

• I=1

UI(x, y, t)ΦI(z) v(x, y, z, t) =

N

• I=1

VI(x, y, t)ΦI(z) w(x, y, z, t) =

M

• I=1

WI(x, y, t)ΨI(z)

z x UN UI UI+1 UI−1 U3 U2 U1 U4 Ith layer 2 1 UI+1 UI UI−1 UI ΦI(z)

N 4 I+1 I I−1 3 2 1 I+1 I I−1

u z

J.N. Reddy

11

Layerwise 2D + 1D

(1a) (2a)

Cubic serendipity element

(in-plane) (through thickness)

Linear Lagrange element Quadratic Lagrange element

(through thickness)

Quadratic serendipity element

(in-plane)

(1b) (2b)

J.N. Reddy

12

Conventional 3D

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Table: Comparison of the number of operations needed to form the element stiffness matrices for equivalent el- ements in the conventional 3-D format and the lay- erwise 2-D format. Full quadrature is used in all. Element Type† Multipli. Addition Assignments 1a (3-D) 1,116,000 677,000 511,000 1b (LWPT) 423,000 370,000 106,000 2a (3-D) 1,182,000 819,000 374,000 2b (LWPT) 284,000 270,000 69,000 † Element 1a: 72 degrees of freedom, 24-node 3-D isopara- metric hexahedron with cubic in-plane interpolation and linear transverse interpolation. Element 1b: 72 degrees of freedom, E12—L1 layerwise element. Element 2a: 81 degrees of freedom, 27-node 3-D isopara- metric hexahedron with quadratic interpolation in all three directions. Element 2b: 81 degrees of freedom, E9—Q1 layerwise el- ement.

13

Layerwise Kinematic Model

3D modeling with 2D & 1D elements

J.N. Reddy

E1 = 25 × 106 psi, E2 = E3 = 106 psi G12 = 0.5×106 psi, G13 = G23 = 0.2×106 psi, ν12 = ν13 = ν23 = 0.25

u(x, a/2, z) =u(a/2, y, z) = 0 v(a/2, y, z) =u(x, a/2, z) = 0 w(x, a, z) =u(a, y, z) = 0

x y

2-D quadratic Lagrangian element three quadratic layers through the thickness

y x z z

a 2 a 2 a 2 a 2 h

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In-plane Stresses predicted by the Layerwise Theory

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8

Inplane normal stress, σ

xx

0.0 0.2 0.4 0.6 0.8 1.0

z/h=0.333 z/h=0.667

Exact 3-D Elasticity Layerwise Mesh 1 Layerwise Mesh 2 CLPT FSDT

90° 0°

23

z/h

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Transverse shear stresses predicted by the Layerwise Theory

• 0.25
• 0.20
• 0.15
• 0.10
• 0.05

0.00

Transverse shear stress, σ

yz

0.0 0.2 0.4 0.6 0.8 1.0

z/h=0.333 z/h=0.667

Exact 3-D Elasticity Layerwise Mesh 1 Layerwise Mesh 2 CLPT (equilibrium) FSDT (equilibrium) FSDT

90° 0°

24

z/h

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uESL

1

(x, y, z) = u0(x, y) + zφx(x, y) uESL

2

(x, y, z) = v0(x, y) + zφy(x, y) uESL

3

(x, y, z) = w0(x, y)

Variable Kinematic Model for Global-Local Analysis

Composite displacement field:

ESL Displacement field: LWT Displacement field:

uLWT

1

(x, y, z) =

N

• I=1

UI(x, y)ΦI(z) uLWT

2

(x, y, z) =

N

• I=1

VI(x, y)ΦI(z) uLWT

3

(x, y, z) =

M

• I=1

WI(x, y)ΨI(z)

ui(x, y, z) = uESL

i

(x, y, z) + uLWT

i

(x, y, z)

25

j1

Slide 17 j1

jnreddy, 6/30/2014

JN Reddy

An Efficient Shell Finite Element

Objective: Develop a robust shell element for the linear and nonlinear analysis of shell structures made of multilayered composites and functionally graded materials that is computationally efficient (i.e., accurate and computationally inexpensive).

( )

2 1

ˆ , 2 2

n k k k k k k k k

h h u u n ψ ξ η ζ ζ

=

  = + + Ψ    



ϕ

F.E. approximation of displacement field 7-parameter displacement field

( )

( )

( ) ( ) ( )

2

ˆ , , , , , 2 2 h h u X u n ξ η ζ ξ η ζ ξ η ζ ξ η = + + Ψ ϕ

Thickness stretch is included

32

• Notable features of the 7-parameter formulation
• Thickness stretching is considered
• Three-dimensional constitutive equations are

used

• Consistent displacement finite element

formulation

• Notable features of present implementation
• Utilization of spectral/ hp finite elem ent

technology to represent the differential geometry and avoid locking

• Static condensation of degrees of freedom

internal to the element

• Applicability to geometrically nonlinear

analysis of FGM and laminated structures

NOTABLE FEATURES

Spectral/hp Finite Element Technology

Improving Numerical Efficiency: Static Condensation

Figure: A high-order spectral/hp finite element discretization (p-level of 4) of a 2-D region: (a) finite element mesh showing elements and nodes and (b) a statically condensed version of the same mesh showing the elements and nodes.

p = 7

28% of the original number of equations

21% of original nonzeros

(b) (a)

Figure: Sparsity patterns for: (a) a high-order finite element mesh and (b) the same high-order mesh using static condensation.

System memory requirements for low-order and high-order problems are similar.

34

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 L = 0.52 m, R = 0.15 m,

h = 0.03 m

 E = 198 x 109 Pa, ν = 0.3  q = 12 x 109 Pa  8 x 1, p = 4

Benchmark Problem 1: Isotropic cylindrical shell subjected to internal pressure

Deformed shape

Thickness deformation vs. axial coordinate

* M. Amabili, “Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells " International *

JN Reddy

Benchmark Problem 2: Pinched cylindrical shell

Finite element solution of deformed mid-surface of pinched cylinder. Deformation magnified by a factor of 5×106 (a) un- deformed shell configuration (b) deformed shell configuration.

(a) (b)

6

3 10 psi, =0.3 E ν = ×

f

4 1.0 lb P  = = 2 600 in, 3 in a R h = = =

Point load P vs. stress, σxx, at point A

Computational time

Elements Nodes Degrees of freedom Time (s) 7-parameter 4 289 2023 66 12-parameter 4 289 3468 473 ANSYS solid 13824 16807 50421 6488 ABAQUS solid 13824 16807 50421 720

JN Reddy

W G Given an operator equation of the form in and in we seek suitable approximation of as . In the least-squares method, we seek the minimum of the sum of squares of the residuals in the appr ( ) ( )

h

A u f B u g u u = =

[ ] [ ]

d d

W G

• ximation
• f the equations:

x

2 2

( ) ( ) ( )

h h h

I u A u f d B u g ds ì ü ï ï ï ï ï ï = =

• +
• í

ý ï ï ï ï ï ï î þ

ò ò 

THE LEAST-SQUARES METHOD

38

JN Reddy

Variational Problem

(based on the least-squares formulation)

[ ] [ ] [ ]

d d d d d d d d d d

W G W G 2 2

( ) ( ) ( ) ( , ) ( ) ( , ) [ ( )] ( ) ( ) ( ) ( ) [

h h h h h h h h h h h h h h

I u A u f d B u g ds u H B u u u u H B u u A u A u d B u B u ds u A ì ü ï ï ï ï ï ï = =

• +
• í

ý ï ï ï ï ï ï î þ Î = Î = + =

ò ò ò ò

 

 

x Thus, the variational problem is to seek such that holds for all where x

[ ]

d

W G

( )] ( )

h h

u f d B u g ds +

ò ò 

x

Fluid Flow (LSFEM) 28

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σ

Γ = ⋅ Γ = Ω = ⋅ ∇ Ω = ∇ + ∇ ⋅ ∇ − ∇ + ∇ ⋅

• n

ˆ ˆ

• n

ˆ in in ] ) ( ) [( Re 1 ) (

u

t σ n u u u f u u u u

T

p

LEAST-SQUARES FORMULATION OF VISCOUS INCOMPRESSIBLE FLUIDS Governing equations (Navier-Stokes equations)

Fluid Flow (LSFEM) 29

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ω

Γ = Γ = Ω = ⋅ ∇ Ω = ⋅ ∇ Ω = × ∇ − Ω = × ∇ − ∇ + ∇ ⋅

• n

ˆ

• n

ˆ in in in in Re 1 ) (

u

ω ω u u ω u u ω f ω u u p

VELOCITY-PRESSURE-VORTICITY FORMULATION

39

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4

Re 10 =

Streamlines Pressure contours Dilatation contours

40

Lid-Driven Cavity Problem

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RESULTS OF OTHER NON-TRIVIAL FLOW PROBLEMS

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Mesh (501 elements; p=4) Close-up of mesh around the cylinder Flow of a Viscous Incompressible Fluid around a Cylinder-1

Fluid Flow (LSFEM) 33

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Fluid Flow (LSFEM) 34

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Robust at moderately high Reynolds numbers: Re = 100 – 104 High p-level solution: p = 4, 6, 8, 10 No filters or stabilization are needed

2D Flows Past a Circular Cylinder-2

Fluid Flow (LSFEM) 35

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Flow of a viscous fluid past a circular cylinder-3

Fluid Flow (LSFEM) 36

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Non-locality can arise from the way we choose to model physical phenomena. Some of the ways the non-locality is modeled are:

• Cosserat or micropolar continuum,
• Strain gradient theories and Modified

couple stress theories,

• Eringen’s integral, differential, and

integro-differential models, and

• Peridynamics, which is an integral

representation of balance laws accounting for long-range forces.

43

JN Reddy

Norm alized bending stiffness increases as the cantilever beam thickness decreases. Measurable at m icron-order thicknesses. (McFarland & Colton, 20 0 5)

3 3 3 3 2

3 3 4 ( 1 for plane stress; 1 for plane strain) PL P EI Ewh k EI L L , δ δ ϕ ϕ ϕ ϕ ν =  = = = = = −

MICRO- AND NANO-ELECTRO- MECHANICAL SYSTEMS

45

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Biomechanics – Bones

Osteons, d = 0.1 or 0.2 mm

Journal of Biomechanical Engineering (1982)

Specimen diameter

• Eff. shear stiffness

Data points Specimen diameter

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LAKE’s USE OF MICROPOLAR THEORY

TO EXPLAIN NONLOCAL EFFECTS

2

ij ij ij kk

σ µε λδ ε = +

( )

2 and are micropolar constants. ( )

ijm m m ij k,k ij i,j j ij ij ij kk ,i

, , , e m k k w f af d b s m e ld e k a b f g gf + +

• =

+ + + =

Classical continuum Cosserat continuum (Cosserats, 190 9) Professor Karan Surana has a presentation

• n som e elem ents of non-classical

continuum m echanics

JN Reddy PA12 with 6% SWNT (showing network formation) introducing distinct microscopic length scale Nematic elastomer with hard nematic phase of random orientation embedded in a soft polymer matrix

Why rotational gradient dependent elasticity? Presence of very stiff secondary phases giving rise to distinct m icroscopic length scale. Intuitively, the secondary phase “rotates” with the m aterial but does not stretch with it; interference between neighbors causes it to resist rotational gradients leading to couple stresses. CNT reinforced polymers and nematic elastomers

STRAIN GRADIENT ELASTICITY THEORY (Srinivasa & Reddy, JMPS, 2013)

JN Reddy

GRADIENT ELASTICITY THEORY

(Srinivasa & Reddy, JMPS) Governing equations in terms of stress resultants : Gradient dependent terms

33 33

, S S E E

ab ab

Ψ Ψ ¶ ¶ = = ¶ ¶

Conventional stress “Drilling” couple stress

, a a

t w Ψ ¶ = ¶

,

T w

ab ab

Ψ ¶ = ¶ “Bending” couple stress

( )

3 33 , , , , ,

N e M N N w

ab b ag b bg ab ab ab ab b a

t d

• =

é ù

• +

= ê ú ë û

JN Reddy

TIMOSHENKO BEAM THEORY

( )

( )

2 2

x L xx xx xz xz zz z y z A xy L

dAdx q w f u dx m + = + +

• +

ò ò ò

dc s de s e s de d d

Nonlocal - 43

( )

,

xx xx x x xx

dM d N f Q dx dx d d d dx w Q N q dx dx

• =
• +

= æ ö ÷ ç

• =

÷ ç ÷ ÷ ç è ø + + M

Mindlin model Srinivasa-Reddy model

2 11 13

2 2

x x

d E d G E dx dx = = + +  q q a l g M M

Needs to be interpreted in the context of a specific problem the square root of the ratio of the moduli of curvature to the shear (a property measuring the effect of the couple stress)

JN Reddy Nonlocal

Microstructure-dependent (gradient elasticity) Mindlin plate

JN Reddy

Bending of a solid circular plate

Clamped Simply-supported Clam ped plate Sim ply-supported plate

Theoretical Bckground

GraFEA: Dependence of nodal force on edge- strains

Typical Element Network with nonlocal forces

Element e Conventional truss with local forces

Khodabakhshi, P, Reddy, J.N., Srinivasa, A.R., 2016. GraFEA: a graph-based finite element approach for the study of damage and fracture in brittle materials, Meccanica, 51 (12): 3129 – 3147.

Capability of GraFEA to Study Fracture

JN Reddy

• Engineers and scientists “m odel” phenom ena

that occurs in nature.

• Continuum m echanics is a m eans to an end; that

is, it provides tools to construct a m athem atical m odel, analyze, and m ake a decision (towards designing and building).

• There is no “com plete” or “exact” m athem atical

m odel of anything we like to m odel & analyze.

• We can only try to “im prove” on what we

already know (often, goal-based thinking).

• Only two things that m atter in engineering: (1)

Reliable functionality (or probability of failure) and (2) cost of the product.

Nonlocal - 4

JN Reddy 49

• Differentiability of field variables is not an

inherent attribute; we endowed them so that we can gain some insights without solving complex problems.

• With the computational tools we have, we can

account for missing terms, or reformulate the classical continuum mechanics with non- classical continuum mechanics (e.g., strain- gradient theories, peridynamics, and others).

JN Reddy

• ●
• ●
• In the end, all numerical

methods involve setting up algebraic relations between the values of the duality pairs (cause and effect) at selected points of the continuum.

• ●
• Typical control

volume,

e

W

Mesh points

• ●

JN Reddy

• Non-classical continuum m echanics brings

additional m eans to address m issing effects from the classical m echanics and explains certain essential m echanism s that are

• bserved in experim ents.
• Eringen’s differential m odel is a diffusion type

stress-gradient m odel. It shows stiffness reduction (flexibility) effect. Thus, it has lim ited application.

• There is experim ental as well as m odeling

evidence that indicates the non-locality in m aterials m anifests in different form s.

JN Reddy

• Generalized (or non-classical) continuum

theories are required to m odel m aterial behavior m ore accurately. Such theories predict reduction in stress concentration factor around holes and cracks, which can give rise to im proved toughness.

• GraFEA has a great potential and it needs to be

developed further for inelastic and ductile m aterials.

• Strain gradient and m odified coupe stress

theories are related, and they show stiffening effect and allow for m ultiple length scales. They can be used to m odel large structures without using full 3-D m odels.

JN Reddy

• We m ust seek physically m eaningful

experim ental validations to understand and predict the risks of failure (i.e., understand what is happening and use it to assess risk of failure).

• Our works m ust be built on sound m echanics

foundation (wisdom to see details).

• We m ust develop robust com putational tools

that m ake use of advances m ade in theoretical developm ents and num erical m ethods.

JN Reddy

I thank you for your interest in my lecture

I thank The Committee on City U Distinguished Lecture Series and Professors C.W. Lim and Q.S. Li That which is not given is lost