Lecture Slides Chapter 5

Practical Stress Analysis with Finite Elements (2nd Edition)

Dr. Bryan J Mac Donald

© Bryan Mac Donald/Glasnevin Publishing 2007‐2011

© Bryan Mac Donald/Glasnevin Publishing 2007-2011

Chapter 5: Material Models Pages 163 to 164 of book

Overview of Material Models

Chapter 3 showed how the implementation of the minimum potential energymethod requires three matrices [S], [B] and [D] which define the behaviour ofthe finite elements.

In Chapter 4 we developed the [S] matrix for each element type andillustrated how to obtain the [B] matrix from differentiation of the [S] matrix.

This chapter completes the specification of element behaviour by establishingthe material property matrix [D]

Each material model will be discussed in increasing order of complexity inreference to table 3.01 in chapter 3



Linear Elastic Isotropic Model

We have already describe this in detail in chapter 2:

• Assumes material behaviour is isotropic – sameproperties in all directions.

• Assumes Hooke’s law is valid: = E• Assumption is only valid until yield stress is

reached – must not be used after this point.

• Useful as a “first guess” even if yield is expected• Requires E and in it’s definition



Linear Elastic Orthotropic Model



Linear Elastic Anisotropic Model

• Linear elastic behaviour• Different elastic constants in every

direction.• i.e. no planes of symmetry• 21 Elastic constants required• Multiple material tests required to

fill [D] matrix with constants.



Non‐Linear Elastic Materials

• Elastomers (rubber like materials)• Foams• Certain polymers• Biomaterials (arterial tissue, collagen, etc.)

• Let’s take rubber as an example:• Up to 500% strain is possible with elastic recovery• The stress‐strain plot is highly non‐linear (i.e. curved)• It has damping properties• Behaviour is time dependant and temperature dependant• Almost incompressible

• See book page 166 for definition of stretch ratio, and the strain invariants:I1 , I2 and I3



Neo‐Hookian Model



Mooney‐Rivlin Models



The Yeoh Model

• Depends only on I1 – less tests required• Inaccurate for small strains• General form and 2 parameter form shown

The Arruda‐Boyce Model

• Also depends only on I1 – less tests required• More accurate for small strains• General form shown



The Gent Model

• Alternative attempt at accurately modelling all strain levels

The Ogden Model

• Allows for up to 700% strain!• The parameter can be varied to allow for strain stiffening

or softening• Has been successfully used to model seals, O‐rings, etc.



The Ogden Compressible Foam Model

The Blatz‐Ko Model

• Both of these are used for rubber foams• See book page 170‐171 for more.



Hyper‐elastic Material Modelling



Visco‐elastic Material Models

• Exhibts both elastic and viscous deformation.• Instantaneous elastic deformation but viscous deformation occurs over time (i.e. creep)• Hysteresis, creep and stress relaxation are all dependant on temperature.• Example materials: Rubber, blood vessels, cartilage, saliva, mucus, Glass, etc.

• Stress relaxation: apply a set strain to a visco‐elastic material and stress will decrease over time!



The Maxwell Visco‐elastic Material Model

• Uses a spring (elastic) and a dashpot (viscous) in series.

• Sudden application of force• Instantaneous deformation of the spring• Followed by creep of dashpot (sloped line)

• Sudden application of displacement• Force in spring instantly rises• Dashpot then slowly expands and relieves

the stress (curved line)



The Kelvin‐Voight Visco‐elastic Material Model

• Uses a spring (elastic) and a dashpot (viscous) in parallel.

• Sudden application of force• Material deforms at a decreasing strain rate

– the slope constantly reduces• Eventually reaches a steady state• If force is removed then relaxes

exponentially to the un‐deformed state• <better than Maxwell!!>

• Sudden application of displacement• Force in spring instantly rises• Followed by instantaneous stress relaxation

due to the dashpot being in parallel.• <Not accurate!! Maxwell is better here.>



The Standard Linear Solid Visco‐elastic Material Model

• Combines Maxwell with a parallel spring.

• Sudden application of force• Instantaneous elastic deformation due to

spring followed by deformation at adecreasing strain rate.

• If force is removed then there is aninstantaneous reaction followed by anexponential decay to the un‐deformed state

• Sudden application of displacement• Force in spring instantly rises• Followed by a gradual stress relaxation due

to the dashpot



Elasto‐Plastic Material Models

• Illustration of strain‐rate effects:

• Modelling the deformation of a bar due to axial compressive load.• In this case the load is applied very quickly in order to model impact of the bar against a

rigid surface.• (a) shows the ¼ symmetry model• (b) shows a model using a strain‐rate dependant material model• (c) shows a model using a strain‐rate independent material model



Strain‐rate Independent Elasto‐Plastic Material Models

• The Bilinear Elasto‐Plastic Mateial Model:

• Two lines: One for the elastic deformation, another for the plastic• E (Young’s Modulus) defines the slope of the elastic line• Etan (Tangent Modulus) defines the slope of the plastic line• Only valid up to the UTS – cannot model stress softening and necking effects.




• Hardening Laws: (Stress range = difference in stress between unloading and next yield event)

Bilinear Kineamtic• (1) Tensile load is applied• (2) Yielding occurs• (3) Plastic deformation• (4) Compressive load applied• (5) Yielding (in compression)• Stress range = 2 Yield Stress

Bilinear Isotropic(1) Tensile Load is applied(2) Yielding occurs(3) Plastic deformation(4) Compressive load applied(5) Yielding (in compression)(6) Stress range = 2 max stress in tension.




• Hardening Laws:




• Multi‐linear Elasto‐Plastic Material Model:

• Obvious extension of the bi‐linear model• Why only have two lines, when you can more closely model the actual stress‐strain curve

with more lines….




• Power Law Elasto‐Plastic Material Model

• Caboche form : combines istotropic and kinematic hardening to give a yield surface that both changes size and moves in stress space.

Basic form:

Isotropic form: Voce hardening law

Kinematic form: uses a back stress for Bauschinger effect




• Power Law Elasto‐Plastic Material Model




• Anisotropic Hill Elasto‐Plastic Material Model



Strain‐rate Dependent Elasto‐Plastic Material Models

• Perzyna visco‐plastic model:

• Pierce visco‐plastic model:

• Cowper‐Symonds model:

• Power law and Cowper‐Symonds:

• Ramsburg‐Osgood model:



Specialised Plasticity Material Models

• Concrete:

• Soil and Granular materials: Drucker‐Prager Model

• Cast Iron:



Specialised Material Models

• Damage Models: Composite Damage model – various modes of damage possibleConcrete Damage model – cracking and crushing

• Shape Memory Alloys:

• Creep: • Foam models : closed cell, viscous, low density, crushable etc.

• Gasket models: high compression and highly non linear



Summary of Chapter 5:

After completing chapter 5, you should:

1. Understand the various material models that are available for use in astructural finite element analysis.

2. Be able to correctly identify the most appropriate model for use in youranalysis.

Lecture Slides Chapter 5

Documents

Transcript of Lecture Slides Chapter 5