High Strain Rate Experiments - ERNETugc/nrcmmpc12/dynamicMechTesting... · High Strain Rate Experiments ... ection and Trasmission of Waves ... Kolsky Bar Experiment - Compression

High Strain Rate Experiments

Krishna JonnalagaddaMechanical Engineering

Indian Institute of Technology Bombay

NRC-M Summer School on

Mechanical Property Characterization

Indian Institute of Science, Bangalore

July 7th, 2012

Krishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 1 / 34

Contents Outline

Topics Covered

Introduction to High Strain Rate Experiments

Review of One Dimensional Elastodynamics

Kolsky Bar Experiments - Compression, Tension, and Torsion

Kolsky Bar Experimental Setup

Wave Propagation Analysis - Dynamic Material Property Extraction

1 Compression and Tension2 Torsion3 Adiabatic Heating

Important Features of Kolsky Bar Experimental Technique

1 Specimen Geometry - Friction, Homogenization, Inertial Effects2 Specimen Material (Metal/Ceramic/Polymer) - Pulse Shaping3 Wave Dispersion4 High/Low Temperature Experiments


High Strain Rate Experiments Introduction

High Strain Rate Experiments on Materials

Dynamic loading causes significant deformation and damage ofmaterials due to the kinetic energy carried by a mass as compared toquasi-static loading

In dynamic loading stress is a function of the particle velocity, i.e., themotion of an individual point mass in the body

Upon impact of a body a stress wave is generated, which travels at aspecified velocity

The velocity of the stress wave is different from the the velocity ofthe particle in motion

Stress wave velocity depends on the nature of the wave and thematerial properties, such as, density and modulus

Dynamic deformation of materials involves stress wave propagation

High strain rate experiments are performed to extract constitutivebehavior, failure, fracture and fragmentation of materials


High Strain Rate Experiments Introduction

High Strain Rate Experimental Techniques

Quasi-static strain rates - < 10−3s−1

Intermediate strain rates - > 10−3s−1

High strain rates - > 102s−1

Very high strain rates - > 104s−1

Ultra high strain rates - > 106s−1

1The choice of experimental technique determines the strain rate rangeaccessible for different materials

Material properties that affect that the strain rate include, density,wave speed, yield strength, etc.

The specimen geometry also changes between the techniques toensure that uniform stress state exists during the experiment

1Ramesh, Experimental Mechanics Handbook, 2009Krishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 4 / 34

High Strain Rate Experiments Review of 1-D Elastodynamics

One Dimensional Wave Propagation in Elastic Media

From Linear Elasticity Theory:

Equillibrium (elasto-statics)

σij ,j + bi = 0 (ΣF = 0)

Equillibrium (elasto-dynamics)

σij ,j + bi = ρui (ΣF = ma)

σij - stress component

bi - body force component

ui - displacement component

ρ - material density




1-D equillibrium in x-direction without body forces:

σxx ,x = ρux (1)

1-D linear elastic material:

σxx = Eεxx (2)

From (1) and (2)

Eεxx ,x = ρux (3)

c20∂2ux∂x2 = ∂2ux

∂t2 (4)

where c0 =√

Eρ is the ” longitudinal wave speed”




General Solution for the 1-D wave equation

c20

∂2ux

∂x2− ∂2ux

∂t2= 0

This is a hyperbolic equation and there are at least two methods to solvethis PDE, separation of variables and method of characteristics.

Using either one of the methods it can be shown that the general solutionis of the form:

u(x , t) = F (x − c0t) + G (x + c0t)

where, F and G are harmonic functions.

F is the Right Traveling Wave (RTW) and G is the Left Traveling Wave(LTW)Krishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 7 / 34



Right Traveling Wave - F (x − c0t). At any time t > 0; x = c0t

u(x , t) = F (x − c0t)

∂u

∂t= −c0F

′(x − c0t);∂u

∂x= F ′(x − c0t)

σ(x , t) = Eεxx = E∂u

∂x= EF ′(x − c0t)

So,

σ(x , t) =−E

c0

∂u

∂t= −ρc0

∂u

∂t

u = ∂u∂t - is the particle velocity

−ρc0 - is the acoustic impedance of the material

if u = ∂u∂t > 0⇒ σ < 0 (Compression)

if ∂u∂t < 0⇒ σ > 0 (Tension)




Left Traveling Wave - G (x + c0t). At any time t > 0; x = −c0t

u(x , t) = G (x + c0t)

∂u

∂t= c0G

′(x + c0t);∂u

∂x= G ′(x + c0t)

σ(x , t) = Eεxx = E∂u

∂x= EG ′(x + c0t)

So,

σ(x , t) =E

c0

∂u

∂t= ρc0

∂u

∂t

if u = ∂u∂t > 0⇒ σ > 0 (Tension)

if u = ∂u∂t < 0⇒ σ < 0 (Compression)



Reflection and Trasmission of Waves

Wave propagation at an interface

There are three waves at theinterface:

Incident σI = fI (x − c1t)

Reflected σR = fR(x + c1t)

Transmitted σT = fT (x − c2t)

Force balance at x = 0:

σI |x=0− + σR |x=0− = σT |x=0+

fI + fR = fT (∗)

Velocity balance at x = 0:

∂uI

∂t|x=0− +

∂uR

∂t|x=0− =

∂uT

∂t|x=0+

−fIρ1c1

+fRρ1c1

=−fTρ2c2

(∗∗)

From (*) and (**) we get,

fR =α− 1

α + 1fI and fT =

2α

α + 1fI

where α =ρ2c2

ρ1c1

is the ” acoustic impedancemismatch”




Free End Reflection:

Incident σI = fI (x − c1t)

Reflected σR = fR(x + c1t)

At the free end, stress is zero

σ|x=0 = fI |x=0 + fR |x=0 = 0⇒ fR = −fI

So, the pulse is of the same shape but of opposite sign, i.e., a compressivepulse reflects as a tensile pulse after interacting with a free surface andvice versa. What about particle velocity?

∂u

∂t=−fIρ1c1

+fRρ1c1

=−2fIρ1c1

The velocity doubles at the free surface




Built in or Rigid End Reflection:

Incident uI = FI (x − c1t)

Reflected uR = FR(x + c1t)

At a rigid end, the boundary conditions are u = 0 and u = 0

ux=0 = FI |x=0 + FR |x=0 = 0⇒ FR = −FI

So, the velocity changes the sign in this case. What about stress?

∂u

∂t=−fIρ1c1

+fRρ1c1

= 0⇒ fI = fR ⇒ σ|x=0 = fI + fR = 2fI

The stress doubles at a rigid end.




Method of Characteristics

Express the 1-D wave equation as c02 ∂ε∂x −

∂u∂t = 0 -(a)

Strain-displacement relationship gives us ∂u∂x −

∂ε∂t = 0 -(b)

(a) + (b) ⇒ c0(c0ε+ u),x −(c0ε+ u),t = 0 -(c)

Now, d(cε+u)dt = ∂(cε+u)

∂xdxdt + ∂(cε+u)

∂t

If dxdt = −c , the above equation is zero and from (c) we get cε+ u =

constant. Similarly (a) - (b) will result in cε− u = 0 for dx/dt = c .


High Strain Rate Experiments Experimental Techniques

Kolsky Bar Experiment - Compression

2

Stress wave is generated by launching a projective onto the input bar

Strains are measured in the input and the output bar

Alignment of the bars and projectile are very important

The bars need to move freely in the supports in their length direction



Kolsky Bar Experiment - Tension

3

A tensile pulse is generated by the sudden release of tensile strainstored in the bar using a clamp

The pulse shape and wave form characteristics are influenced by theclamp design

Finite element analysis is required for specimen design3Ramesh, Experimental Mechanics Handbook, 2009



Tension/Compression x - t Diagram

4

Lagrangian or x-t diagrams are extremely useful in the analysis ofKolsky bar experiments



Kolsky Bar Experiment - Torsion

5

In a torsion bar a torsional (shear) wave is propagated in the bars andthe specimen

Torsional bars do not required dispersion correction

Large strains can be generated in a torsional bar

Bending waves need to avoided during operation

Finite element analysis is required for the specimen design5Ramesh, Experimental Mechanics Handbook, 2009



Kolsky Bar Experiment - Torsion - x-t Diagram

6

6ASM Metals Handbook, Vol. 8Krishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 18 / 34

High Strain Rate Experiments Experimental Analysis

Kolsky Bar Analysis - Compression and Tension

A0 - Initial specimen cross section area and L0 - Initial specimen lengthAverage Stress in the specimen:

σ(t) =P1(t) + P2(t)

2A0

P1(t) = EAε1 = EA(εI (t) + εR(t)) and P2(t) = EAε2 = EAεT (t)

If the specimen stress state is homogeneous (equillibrium)

P1 = P2 and εI + εR = εT

Therefore, engineering stress and true stress are:

σ(t) =EAεT (t)

A0and σt(t) = σ(t)[1− ε(t)]



Kolsky Bar Experiments - Analysis

Compression and TensionNow to calculate strain in the specimen, the average strain rate:

ε(t) =u2 − u1

L0

u1 = uI (t) + uR(t) =−σI

ρc0+σR

ρc0=−EεI + EεR

ρc0and u2 =

−EεTρc0

But due to homogenization, εI + εR = εT . So, engineering strain rate andtrue strain rate are given by

ε =2EεR(t)

ρc0L0=

2c0εR(t)

L0and ε(t) =

ε(t)

1− ε(t)

Then the engineering strain and true strain are calculated using,

ε(t) =

∫0

t

ε(τ)dτ and ε(t) = −ln[1− ε(t)]



Kolsky Bar Experiments - Compression Analysis

7



Kolsky Bar Experiments - Compression Analysis

8



Kolsky Bar Experiments - Torsion Bar Analysis

9

The torque, T and the angular velocity, θ, of the bar for a given density ρand polar moment of inertia Jc are related by:

T = ρJc θ

Maximum torque in a bar is given by:

Tmax =τyJc

r⇒ θ =

τy

r√ρG

The average shear strain rate γs(t) and shear strain are given by:

γs(t) =rsLs

[θ1 − θ2] and γs(t) =

∫0

t

γs(τ)dτ

9ASM Metals Handbook, Vol 8Krishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 23 / 34



The relative change in angular velocity is given by (with one strain gage):

[θ1 − θ2] =2[−TR(t)]

ρJcand TR(t) =

GJcγR(t)

rb

Now to compute the shear stress we use to the strain measured in thetransmitted bar, γT assuming that the stress in the specimen is uniform ⇒torque at both ends of the specimen is same

τs(t) =Ts(t)

2πrs2ts=

Grb3

4rs2tsγT (t)

where, rs is the mean radius of an hollow specimen, rb is the radius of thetorsion bars, ts is the thickness of the specimen, TR(t) torque calculatedfrom the reflected strain. 10




Torsion Strain, Strain Rate and Stress vs. Strain Curves

11

In the torsional Kolsky bar experiments static compression andtension loads can be superimposed

Due to the long pulses generated in the torsional bars the bendingand axial forces need to be avoided

Specimen gripping is critical as any misfit arrangement can cause longrise times and errors in strain measurement.



Kolsky Bar Experiments - Adiabatic Heating

Under dynamic loading, inelastic deformation causes temperature risein the specimen

The heat removal rate in a material depends on the thermaldiffusivity α (= 1.14cm2s−1 for copper).

The distance the heat travels is given by 2√αt

At higher strain rates, the heat generated from the specimen duringentire experiment is accumulated - this is called adiabatic heating

Temperature rise during adiabatic heating is given by

∆T =β

ρCp

∫ εf

0σdε

where β depends on the plastic work converted into heat

Implications of this adiabatic heating include thermal softening andshear banding


High Strain Rate Experiments Experimental Setup

Kolsky Bar Experiment - Design

Kolsky bar design has three important ratios to consider but mostimportantly the specimen size, i.e., for a cylindrical sample the lengthl0 and diameter d0 and the projectile velocity V are chosen first.

The ratios are:1 L/D - Bar length to bar diameter ratio - typically about 1002 D/d0 - Bar diameter to specimen diameter - about 2 to 43 l0/d0 - Specimen length to its diameter - about 0.5 to 1

The main components of Kolsky bar setup are:1 Projectile launcher - e.g., compressed air gas gun2 Input and output bars - e.g., maraging steel bars3 Strain gauges to measure strain in the bars - e.g., foil gages4 Alignment fixture to keep the bars straight and concentric5 Data acquisition with required bandwidth and resolution - e.g.,

oscilloscope

The specimen size chosen should represent the bulk response of thematerials



Kolsky Bar Experiment - Gas Gun

Design parameters include:Projectile mass

Projectile velocity

Pressure in the breach chamber



Kolsky Bar Experiment - Strain Measurement

Quarter Bridge vs. Half Bridge - Depends on length of the bar andoutput signal required

Bonded Foil vs. Semiconductor Strain Gages - Depends on gagefactor required

Signal Amplification is necessary when amplitude is small

Data acquisition should have enough bandwidth to capture sharpgradients in the signal


High Strain Rate Experiments Comments

Kolsky Bar Experiment - Specimen Geometry

Considerations for specimen geometry and preparation

Specimen surfaces need to be flat, parallel and smooth forcompression experiments

Specimen should not have stress concentration regions for allcompression, tension and torsion

In the case of tension and torsion finite element calculations need tobe performed with the end conditions

While testing ceramic specimens the punch effect should be avoided

For ceramic specimens pulse shaping may be required to allowuniform loading before critical strength is reached

For very soft materials lower stiffness bars are required to measurestress

For tension and torsion experiments local strain measurement may berequired for accurate stress vs. strain curves

Strain control by varying pulse width or by using a collarKrishna Jonnalagadda (IIT Bombay) NRC-M Summer School - June 2012 July 7th, 2012 30 / 34

High Strain Rate Experiments Comments

Kolsky Bar Experiment - Requirements

Force equilibrium ⇒ P1 = P2

Stress equilibration is required for output bar measurement torepresent average stress in the specimen.

Friction Effects - will lead to specimen barreling and uniaxial stressstate is not maintained

Friction can be removed by ensuring flat and smooth bar andspecimen surfaces and by the use of lubricants

Dispersion Effects - Longitudinal waves suffer from geometricdispersion, e.g., due to Poisson expansion or contraction.

Dispersion effects the strain pulses, superimposes oscillations in thespecimen loading and shorter pulses cause much worse dispersioneffects.

Dispersion effects can be removed by corrections performed on thestrain signals or by introducing pulse shaping


High Strain Rate Experiments Some Results

Dynamic Compression Kolsky Bar

Projectile

Vertical Posts Input Bar Output Bar Momentum Trap Specimen

12

Desktop Kolsky bar designed to load specimens at very high strainrates of 104 to 105

The typical projectile velocity ranges from 10-50 m/s12Jonnalagadda et al., MSE-A, submitted


High Strain Rate Experiments Some Results

Dynamic Strength of Armor Materials

Dynamic strength, failure, rate-dependence and fracture in metals andceramicsTemperature-Strain Rate dependence in materialsPhenomelogical and physical models such as Johnson & Cook,Zerilli-Armstrong (Z-A) and Mechanical Threshold Stress (MTS) formaterial behavior


High Strain Rate Experiments References

References

1 D. Jia and K. T. Ramesh, ”A rigorous assessment of the benefits ofminiaturization in the Kolsky bar system,” Experimental Mechanics,2004.

2 K. T. Ramesh, ”High Strain Rate and Impact Experiments,”Experimental Mechanics Handbook, Springer, 2009.

3 Mechanical Testing and Evaluation, ASM Handbook, Vol 8, 2000.

4 B. A. Gama, et al., ”Hopkinson Bar Experimental Technique: ACritical Review,”Appl Mechanics Reviews, vol 57, 2004.

5 M. A. Meyers, ”Dynamic Behavior of Materials, John Wiley and Sons,1994.

6 G. T. Gary, ”Classic Split Hopkinson Pressure Bar Testing,” ASMInternational, 2000


Documents

High Strain Rate Experiments - ERNETugc/nrcmmpc12/dynamicMechTesting... · High Strain Rate Experiments ... ection and Trasmission of Waves ... Kolsky Bar Experiment - Compression