vignan chapter 8 sources of nonlinearity

SOURCES OF NONLINEARITIES

Geometric

Material

Force Boundary Conditions

Displacement Boundary Conditions

Prof .N. Siva Prasad, Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras

GEOMETRIC NONLINEARITY Physical source

Change in geometry as the structure deforms is taken into account in setting up the strain displacementand equilibrium equations.

Applications

1.Slender structures in aerospace, civil and mechanical engineering applications. 2.Tensile structures such as cables and inflatable membranes. 3.Metal and plastic forming. 4.Stability analysis of all types.


GEOMETRIC NONLINEARITY CONTD..

Mathematical sourceStrain-displacement equations:

e = Du (2.1) The operator D is nonlinear when finite strains (as opposed to

infinitesimal strains) are expressed in terms of displacements.

Internal equilibrium equations:b = −D∗σ (2.2)

In the classical linear theory of elasticity, D = ∗ DT is the formal adjoint of D, but that is not necessarily true if geometric nonlinearities are considered.


GEOMETRIC NONLINEARITY CONTD.. Large strain

The strains themselves may be large, say over 5%. Ex: rubber structures (tires, membranes)

Small strains but finite displacements and/or rotations. Slender

structures undergoing finite displacements rotations

although the deformational strains may be treated as infinitesimal. Example: cables, springs


Linearized prebucking.When both strains and displacements may be

treated as infinitesimal before loss of stability by buckling.These may be viewed as initially stressed members.

Example: many civil engineering structures such as buildings and stiff (non-suspended) bridges.

GEOMETRIC NONLINEARITY CONTD..


MATERIAL NONLINEARITY Physical source

Material behavior depends on current deformation state and possibly past history of the deformation.

Other constitutive variables (prestress, temperature, time, moisture, electromagnetic fields, etc.) may be involved.

ApplicationsStructures undergoing

nonlinear elasticity plasticity viscoelasticity creep, or inelastic rate effects.


MATERIAL NONLINEARITY CONTD..

Mathematical source The constitutive equations that relate stresses and strains.

For a linear elastic material σ = Ee where the matrix E contains elastic moduli.

Note: If the material does not fit the elastic model, generalizations of this equation are necessary, and a whole branch of continuum mechanics is devoted to the formulation, study and validation of constitutive equations.


MATERIAL NONLINEARITY CONTD..The engineering significance of material nonlinearities varies

greatly across disciplines. civil engineering

deals with inherently nonlinear materials such as concrete, soils and low-strength steel.

mechanical engineeringcreep and plasticity are most important, frequently

occurring in combination with strain-rate and thermal effects.

aerospace engineeringmaterial nonlinearities are less important and tend to

be local in nature (for example, cracking and “localization” failures of composite

materials).


Material nonlinearities may give rise to very complex phenomena such as path dependence, hysteresis, localization, shakedown, fatigue, progressive failure. The detailed numerical simulation of these phenomena in three dimensions is still beyond the capabilities of the most powerful computers.


MATERIAL NONLINEARITY CONTD..

FORCE BC NONLINEARITY Physical Source

Applied forces depend on deformation.

ApplicationsThe most important engineering application concerns pressure loads of fluids.

Ex:1. Hydrostatic loads on submerged or container

structures; 2. Aerodynamic and hydrodynamic loads caused by

the motion of aeriform and hydro form fluids (wind loads, wave loads, and drag forces).


DISPLACEMENT BC NONLINEARITY Physical source

Displacement boundary conditions depend on the deformation of the structure.

ApplicationsThe most important application is the contact problem,

in which no-interpenetration conditions are enforced on flexible bodies while the extent of the contact area is unknown.

Non-structural applications of this problem pertain to the moregeneral class of free boundary problems,

example: ice melting, phase changes, flow in porous media. The determination of the essential boundary conditions is a key part of the solution process.


Some solution methodFor a time independent problem[K]{D}={F}

For a linear analysis [K] and {R} are independent of [D].For nonlinear analysis [K] and {R} are regardedas function of {D}

Consider [K] is a function of {D} and can be computedfor a given {D} Consider a nonlinear spring in Fig 1

Spring stiffness [K]=K0+KN

K0=constant termKN=depends on deformation

(K0+KN)u=P

Where u=displacementP=load And KN=f(u) and depends on [D]

Note: 1.when KN is known in terms of u,P can be calculated in terms of u2.Explicit solution for u is not available3.u can be determined by iterative methods

K U

P

u

PHardeningKN>0

SofteningKN<0

KN=0

Fig 1

13

10

APu K

20 1( ( ))

APu K f u

1 1 11 0 2 0 1 1 0, ( ) ,......, ( )A N A i Ni Au k p u k k p u k k p

Direct substitution

Let KN<0 (softening springPA is the load applied

Assume KN=0 first iterationUA=displacement produced for the first iteration

Use u1 to compute the new stiffness. K0+KN1=K0+f(u)

Writing symbolically

This calculations are interpreted graphically in Fig 2.2

Note:1.Approximate stiffness K0+KNi can be regarded as secants of the actual curve2.After several iterations, the secant stiffness=K0+KN

3.stiffness=PA/UA u=uA is closely approximated

u

P

u1 u2 u3

12

3

a b c

K0

Slope=K0-KN1

14

Alternative methodTake KN is to the right hand side

1 1 11 0 2 0 1 1 1 0, ( ),......, ( )A A N i A Ni iu k p u k p k u u k p k u

u

P

u1 u2 uA

1

2A

a b

K0

KN1u1=pa-p1

KN2u2=p1-p2

p1

pA

KN1u1

KN1u1

KN2u2

15

N-R solution Modified N-R solution

uu1 u2 uB

12

a b B

uA

PA

PB

P1

PB- PA

PB- PA

Δu1 Δu2

P

(Kt)A

(K1)1

uu1 u2 uB

12

a b B

uA

PA

PB

P1

PB- PA

PB- PA

Δu1 Δu2

P

(Kt)1

(K1)1


Total Lagrangian FormulationThree kinematic descriptions of geometrically nonlinear finite element analysis

are in current use in programs that solve nonlinear structural problems. They an

be distinguished by the choice of reference configuration.

1. Total Lagrangian description (TL). The reference configuration is seldom or never

changed: often it is kept equal to the base configuration throughout the analysis.

Strains and stresses are measured with respect to this configuration.

2. Updated Lagrangian description (UL). The last target configuration, once reached,

becomes the next reference configuration. Strains and stresses are redefined as

soon as the reference configuration is updated.

3. Corotational description (CR). The reference configuration is split. Strains and

stresses are measured from the corotated configuration whereas the base

configuration is maintained as reference for measuring rigid body motions.

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Remark : The TL formulation remains the most widely used in continuum-based finite element

codes. The CR formulation is gaining in popularity for structural elements such as beams,

plates and shells. The UL formulation is primarily used in treatments of vary large strains

and flow-like behavior.

Coordinate Systems : Configurations taken by a body or element during the response analysis

are linked by a Cartesian global frame, to which all computations are ultimately referred.

There are actually two such frames:

(i) The material global frame with axes {X} or {X, Y, Z}.

(ii) The spatial global frame with axes {x } or {x, y, z}.

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The material frame tracks the base configuration whereas the spatial frame tracks all

others. This distinction agrees with the usual conventions of classical continuum

mechanics. In the present work both frames are taken to be identical, as nothing is

gained by separating them. Thus only one set of global axes, with dual labels, is drawn

in Figure 1. In stark contrast to global frame uniqueness, the presence of elements

means there are many local frames to keep track of. More precisely, each element is

endowed with two local Cartesian frames:

(iii) The element base frame with axes { X } or { X, Y, Z}.

(iv) The element reference frame with axes {x} or {¯x, ¯y, ¯z}.

The base frame is attached to the base configuration. It remains fixed if the base is fixed.

It is chosen according to usual FEM practices.

18Prof .N. Siva Prasad, Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras

There are two ways to construct TL elements:

1. The Standard Formulation (SF)

2. The Core Congruential Formulation (CCF).

The first method is easier to describe and will be presented in this Chapter through examples.

The second one is more flexible and powerful but it is more dif cult to teach because it

proceeds in stages.

19Prof .N. Siva Prasad, Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras

vignan chapter 8 sources of nonlinearity

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