JN Reddy

J. N. ReddyCenter for Innovations in Mechanics for Design

and ManufacturingTexas A&M University, College Station, Texas

jnreddy@tamu.edu; http://mechanics.tamu.edu

City U Distinguished LectureCity 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

JN Reddy

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-complementary variationalprinciples in mathematical 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 andT ET u f+ = Ω* ( ) 0 inS CS c h+ = Ω

All conventional as well as mixed variational principles are derived. Several of these principles formed the basis of the mixed, hybrid, and assumed strain finite element models (they were not cited often because our work was a bit mathematical and buried in the literature).

8

JN Reddy

( , , )xz x y zs

z

55 45( , , )

;

xz xz yz

xz yz

x y z Q Qu w v wz x z y

s g g

g g

= +∂ ∂ ∂ ∂= + = +∂ ∂ ∂ ∂

Transverse shear stress

2 30

2 30

0

( , , )( , ,

( , ) ( , ) ()

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

y

x

y y

x xx y x y x y x yx

u x y z u z z zv x y z v z z zy xw

y x y xx y z w

yx y

f q l

f q l

= + + +

= + + +

=

Displacement field

J.N. Reddy, “A simple higher-order theory for laminated composite plates,” J. of Applied Mechanics, 51, 745-752 (1984). (over 2000 citations)

J.N. Reddy and C.F. Liu, “A higher-order shear deformation theory for laminated elastic shells,” Int.J. of Engng. Sci., 23(3), 319-330 (1985). (800 citations)

CLPT FSDT

TSDT

12

JN Reddy

xzs

z2

2

( , ) 3 ( , )

( , ) 3

( , )

( ( ), ) ,

2

2

xxz

y yz

x

y

x

y

wx y z x yxwx y z

u w x yz xv w x yz

z

z yy

xy

g

g q

f l

l

q

f

+ +∂ ∂= + =∂ ∂∂ ∂= + =

∂+∂∂+

∂ ∂+

∂+

Transverse shear strains

Vanishing of transverse shear stresses on the bounding planes

2 2

( , , / 2) ( , , / 2) 0 ( , ) ( , ) 04 4( , ) ( , ) , ( , ) ( , )

3 3

y

x x y y

xz yz xx y h x y h x y x yw wx y x y x y x y

h x h yl

s s

f f

q

l

q± = ± = = =

∂ ∂ = − + = − + ∂ ∂

0

0

0

3

2

3

2

43

( , , )

( , , )

(

( , ) ( , )

( , ) ( , )

,3

), ( ,

4

)

xx

y y

u x y z u z

v x y z v z

w x y z w

zx y x y

x y

wh xz wx y

x yh y

f

f

f

f

= +

=

∂ − +

+

=

∂ ∂ − + ∂

∂x∂w0

φx

(u,w)

u0 w0( , )

TSDT

JN Reddy

0 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

Def

lect

ion,

w _ TSDT

3-D Elasticity Solution

FSDTCLPT

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=0.5E2G23=0.2E2 , n12=0.25

E2 = 106 psi (7 GPa)3D

CLPT

JN Reddy

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

JN Reddy

Equilibrium of interlaminarstresses

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+1σzx

(k+1)

σzy(k+1) σzz

(k+1)

y

x

z

9

1k kxx xx

yy yy

xy xy

s ss ss s

+ ≠

JN Reddy

Single-Layer Theories

Equilibrium Requirements

10

1 1k k k kzz zz zz zz

yz yz yz yz

xz xz xz xz

s s e es s e es s e e

+ + = ≠

( ) ( 1)because k kij ijQ Q +≠

kth layer (k+1)th layer

y

x

z

1 1

,

k k k kxx xx zz zz

yy yy yz yz

xy xy xz xz

e e e ee e e ee 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−1u

z

J.N. Reddy 11

Layerwise 2D + 1D

(1a)

(2a)

Cubicserendipity

element

(in-plane) (through thickness)

LinearLagrangeelement

QuadraticLagrangeelement

(through thickness)

Quadraticserendipity

element

(in-plane)

(1b)

(2b)J.N. Reddy 12

Conventional 3D

JN Reddy

Table: Comparison of the number of operations needed toform 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) 1,116,000 677,000 511,0001b (LWPT) 423,000 370,000 106,000

2a (3-D) 1,182,000 819,000 374,0002b (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 andlinear transverse interpolation.Element 1b: 72 degrees of freedom, E12—L1 layerwiseelement.Element 2a: 81 degrees of freedom, 27-node 3-D isopara-metric hexahedron with quadratic interpolation in allthree directions.Element 2b: 81 degrees of freedom, E9—Q1 layerwise el-ement.

13

Layerwise Kinematic Model3D 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 quadraticLagrangian element

three quadratic layersthrough the thickness

y

x

z

z

a 2 a 2

a 2

a 2

h

22

JN Reddy

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 ElasticityLayerwise Mesh 1Layerwise Mesh 2CLPTFSDT

90°

0°

23

z/h

JN Reddy

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 ElasticityLayerwise Mesh 1Layerwise Mesh 2CLPT (equilibrium)FSDT (equilibrium)FSDT90°

0°

24

z/h

JN Reddy

uESL1 (x, y, z) = u0(x, y) + zφx(x, y)

uESL2 (x, y, z) = v0(x, y) + zφy(x, y)

uESL3 (x, y, z) = w0(x, y)

Variable Kinematic Modelfor Global-Local Analysis

Composite displacement field:

ESL Displacement field:

LWT Displacement field:

uLWT1 (x, y, z) =

N

I=1

UI(x, y)ΦI(z)

uLWT2 (x, y, z) =

N

I=1

VI(x, y)ΦI(z)

uLWT3 (x, y, z) =

M

I=1

WI(x, y)ΨI(z)

ui(x, y, z) = uESLi (x, y, z) + uLWT

i (x, y, z)

25

j1

Slide 17

j1 jnreddy, 6/30/2014

JN Reddy

An Efficient Shell Finite ElementObjective: Develop a robust shell element for thelinear and nonlinear analysis of shell structuresmade of multilayered composites and functionallygraded materials that is computationally efficient(i.e., accurate and computationally inexpensive).

( ) 2

1ˆ,

2 2

nk k k kk k

kk

h hu u nψ ξ η ζ ζ=

= + + Ψ ϕ

F.E. approximation of displacement field

7-parameter displacement field

( )( ) ( ) ( ) ( )2 ˆ, , , , ,2 2h hu X u nξ η ζ ξ η ζ ξ η ζ ξ η= + + Ψϕ

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 element

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 TechnologyImproving Numerical Efficiency: Static Condensation

Figure: A high-order spectral/hp finiteelement discretization (p-level of 4) of a2-D region: (a) finite element meshshowing elements and nodes and (b) astatically condensed version of the samemesh 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) ahigh-order finite element mesh and(b) the same high-order mesh usingstatic condensation.

System memory requirementsfor low-order and high-orderproblems are similar. 34

JN Reddy

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 pinchedcylinder. Deformation magnified by a factor of 5×106 (a) un-deformed shell configuration (b) deformed shell configuration.

(a) (b)

63 10 psi, =0.3E ν= ×

f4 1.0 lbP = =2 600 in, 3 ina R h= = =

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

Computational timeElements Nodes Degrees of

freedom Time (s)

7-parameter 4 289 2023 66

12-parameter 4 289 3468 473

ANSYS solid 13824 16807 50421 6488

ABAQUSsolid 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 dW G

oximation

of the equations:

x22

0 ( ) ( ) ( )h h hI u A u f d B u g dsì üï ïï ïï ï= = - + -í ýï ïï ïï ïî þò 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

220 ( ) ( ) ( )

( , ) ( )

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

( ) [

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 üï ïï ïï ï= = - + -í ýï ïï ïï ïî þÎ

= Î

= +

=

ò ò

ò ò

x

Thus, the variational problem is to seek such thatholds for all

where

x

[ ]dW G

( )] ( )h hu f d B u g ds+ò òx

Fluid Flow (LSFEM) 28

JN Reddy

σΓ=⋅

Γ=

Ω=⋅∇

Ω=∇+∇⋅∇−∇+∇⋅

onˆˆ

onˆ

in0

in])()[(Re1)(

u

tσn

uu

u

fuuuu Tp

LEAST-SQUARES FORMULATIONOF VISCOUS INCOMPRESSIBLE FLUIDS

Governing equations (Navier-Stokes equations)

Fluid Flow (LSFEM) 29

JN Reddy

ωΓ=

Γ=

Ω=⋅∇

Ω=⋅∇

Ω=×∇−

Ω=×∇−∇+∇⋅

onˆ

onˆ

in0

in0

in

inRe1)(

u

ωω

uu

ω

u

0u ω

fωuu p

VELOCITY-PRESSURE-VORTICITY FORMULATION

39

JN Reddy

4Re 10=

Streamlines

Pressure contours Dilatation contours

40

Lid-Driven Cavity Problem

JN Reddy

RESULTS OF OTHER NON-TRIVIAL FLOW PROBLEMS

JN Reddy

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

JN Reddy Fluid Flow (LSFEM) 34

JN Reddy

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

JN Reddy

Flow of a viscous fluid past a circular cylinder-3

Fluid Flow (LSFEM) 36

JN Reddy

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

Normalized bending stiffness increases as the cantilever beam thickness decreases. Measurable at micron-order thicknesses. (McFarland & Colton, 2005) 3 3

3 3

2

33 4

( 1 for plane stress; 1 for plane strain)

PL P EI EwhkEI L L

,

δδ ϕ ϕ

ϕ ϕ ν

= = = =

= = −

MICRO- AND NANO-ELECTRO-MECHANICAL SYSTEMS

45

JN Reddy

Biomechanics – Bones

Osteons, d = 0.1 or 0.2 mm

Journal of Biomechanical Engineering (1982)

Specimen diameter

Eff.

shea

r stif

fnes

s

Data points

Specimen diameter

JN Reddy

LAKE’s USE OF MICROPOLAR THEORYTO EXPLAIN NONLOCAL EFFECTS

2ij ij ij kkσ μ ε λδ ε= +

( )2

and are micropolar constants.

( )ijm m m

ij k,k ij i, j j

ij ij ij kk

,i

, , ,

em

k k w f

af d b

s m e ld e

k a b

f

g

gf

+ + -

=

+

+ +

=

Classical continuum

Cosserat continuum (Cosserats, 1909)

Professor Karan Surana has a presentation on some elements of non-classical continuum mechanics

JN Reddy

PA12 with 6% SWNT (showing network formation) introducing distinct microscopic length scale

Nematic elastomer with hard nematicphase of random orientation embedded in

a soft polymer matrix

Why rotational gradient dependent elasticity? Presence of very stiff secondary phases giving rise to distinct microscopic length scale. Intuitively, the secondary phase “rotates” with the material 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 :

Gradientdependent terms

3333

,S SE Eab

ab

Ψ Ψ¶ ¶= =

¶ ¶Conventional stress

“Drilling” couple stress

,a

a

twض

=¶

,

Twab

ab

Ψ¶=

¶

“Bending” couple stress

( )3

33

0

0

, ,

, , ,

N e

M N N wab b ag b bg

ab ab ab ab b a

t

d

- =

é ù- + =ê úë û

JN Reddy

TIMOSHENKO BEAM THEORY

( )

( )

0

0

0 2 2 x

L

xx xx xz xz zz z yzA xy

L

dAdx

q w f u dx

m+= + +

- +

ò ò

ò

dcs de s e s de

d d

Nonlocal - 43

( ) 0 0

0

, xxxx x

x xx

dMd N f Qdx dx

d d ddx

wQ N qdx dx

- - = - + =

æ ö÷ç- - =÷ç ÷÷çè ø++M

Mindlin model Srinivasa-Reddy model2

11 132 2x xd E dG Edx dx

= = ++q q

a lgM 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 plateClamped

Simply-supported

Clampedplate

Simply-supportedplate

Theoretical Bckground

GraFEA: Dependence of nodal force on edge- strains

Typical Element

Network withnonlocal forces

Element eConventional trusswith 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 “model” phenomena that occurs in nature.

Continuum mechanics is a means to an end; that is, it provides tools to construct a mathematical model, analyze, and make a decision (towards designing and building).

There is no “complete” or “exact” mathematical model of anything we like to model & analyze.

We can only try to “improve” on what we already know (often, goal-based thinking).

Only two things that matter 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, eW

Meshpoints

●

●

● ●

●●●

● ●●●

●

●●

●●

●

●●●

● ● ●

●●●●●

●

●●

●●●

●●

JN Reddy

Non-classical continuum mechanics brings additional means to address missing effects from the classical mechanics and explains certain essential mechanisms that are observed in experiments.

Eringen’s differential model is a diffusion type stress-gradient model. It shows stiffness reduction (flexibility) effect. Thus, it has limited application.

There is experimental as well as modeling evidence that indicates the non-locality in materials manifests in different forms.

JN Reddy

Generalized (or non-classical) continuum theories are required to model material behavior more accurately. Such theories predict reduction in stress concentration factor around holes and cracks, which can give rise to improved toughness.

GraFEA has a great potential and it needs to be developed further for inelastic and ductile materials.

Strain gradient and modified coupe stress theories are related, and they show stiffening effect and allow for multiple length scales. They can be used to model large structures without using full 3-D models.

JN Reddy

We must seek physically meaningful experimental 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 must be built on sound mechanics foundation (wisdom to see details).

We must develop robust computational tools that make use of advances made in theoretical developments and numerical methods.

JN Reddy

I thank you for your interest in my lecture

I thank The Committee on

City U Distinguished Lecture Seriesand

Professors C.W. Lim and Q.S. Li

That which is not given is lost