THERMAL EFFECTS IN DYNAMIC PLASTICITY, NUMERICAL SOLUTION AND EXPERIMENTAL INVESTIGATIONS (THERMOGRAPHIC-INFRARED DETECTION)

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THERMAL EFFECTS IN DYNAMIC PLASTICITY, NUMERICAL SOLUTION AND EXPERIMENTAL INVESTIGATIONS (THERMOGRAPHIC-INFRARED

DETECTION)

W. Nowacki

To cite this version:

W. Nowacki. THERMAL EFFECTS IN DYNAMIC PLASTICITY, NUMERICAL SOLUTION AND

EXPERIMENTAL INVESTIGATIONS (THERMOGRAPHIC-INFRARED DETECTION). Journal

de Physique Colloques, 1985, 46 (C5), pp.C5-113-C5-119. �10.1051/jphyscol:1985514�. �jpa-00224744�

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JOURNAL DE PHYSIQUE

Colloque C5, supplkment au n08, Tome 46, aoQt 1985 page C5-113

T H E R M A L EFFECTS IN DYNAMIC PLASTICITY, NUMERICAL SOLUTION AND E X P E R I M E N T A L I N V E S T I G A T I O N S (THERMOGRAPH I C-I N F R A R E D D E T E C T I O N )

W.K. Nowacki

I n s t i t u t e of FundarnentaZ TechnoZogical Research of the Polish Academy of Sciences, 00 049 Warszawa, uZ. Swietokrzyska 21, Poland

Rksumb - On btudie l a solution du problkme de I'impact d'une tige blastoviscoplastique s u r une p a r o i rigide,en supposant que l e processus est localement adiabatique e i dans le cas des grandes dbformations. On tient compte de I'influence de la tempbrature se produisant l o r s du choc s u r l e champ des dbformations. On a effectub les expbriences

correspondantes en u t i l i s a n t un systkme de b a r r a s dlHopkinson et une camgt-a i n f r a - rouge pyroblectrique. On compare les r k s u l t a t s des calculs numbriques avec les donnb-

es expkrimentales.

Abstract - The problem of the impact of a r i g i d wall by an elastoviscoplastic r o d i s examined under the assumption that the process i s l o c a l l y adiabatic and i n the case of f i n i t e deformations.The influence of the temperature produced by the impact on the deformation f i e l d in the r o d i s discussed. Experimental investigation i s performed i n the Hopkinson p r e s s u r e b a r system and a thermovision set. The r e s u l t of numerical com- putations a r e compared with experimental data.

I

-

INTRODUCTION

I t i s discussed how the temperature f i e l d depends on the viscoplastic deformation i n the dynamically deformed b a r under the assumption that the process i s locally adiabatic.

Solution i s found t o the problem of wave propagation i n an elasticviscoplastic b a r which s t r i - kes against an undeformable obstacle. The theory of f i n i t e deformations of thermoelastovisco- p l a s t i c i t y of J.Mandel /2,3/ i s used. The effect of tranverse i n e r t i a i s disregarded,but that of transverse deformation i s taken into account because i t may be important i n case of f i n i t e deformation.

I I

-

FORMULATION OF T H E THERMOMECHAN I CAL PROBLEM

We summarise h e r e the principal r e s u l t s from the theory of elastic-viscoplastic continua as developed by J.Mandel /2,3/ and also obtain some related r e s u l t s f o r l a t e r use.

The gradient o f total transformation is:

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- ^{F = E P}

where _E and _Pare the gradients of elastic and plastic transformation rrspartivrly.Thegrad- ient of velocity of the displacement is:

(2) 1

V = g r a d ~ = i E - l

= E L -

+ _ E ~ , l I - l

-

where

ye = g g1

^and

f

⁼_E

_p-l

E-I a r e the elastic and plastic parts, e -

The velocity of elastic deformation D = $(

ve + $7

can by w r i t t e n i n terms of t h r r a t e of Cauchy s t r e s s tensor and the temperature i n the form

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985514

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JOURNAL DE PHYSIQUE

where T i s the temperature,

2

i s the Truesdell d e r i v a t i v e of the stress t e n s o r , C and _A - elasticity coefficients depending on the i n i t i a l configuration:

-1 -1 0 -1 -1 -1 0 -1

(4) Lmnrs= Eim E j n L i j h l E h r E ~ s 3 Amn=Eim A.. Ejn,

0 . 0 l J

where L . I S a matrix of e l a s t i c i t y coefficients and A..

-

matrix of thermic dilatation coefficients, lJh; i s the mass density of the actual configuratf$n.

I n the elastic-viscoplastic medium w i t h instantaneous deformations,the gradient of r a t e of plas- t i c transformation i s the sum of the instantaneous p l a s t i c i t y term and viscoplasti-

c i t y term as a function of actual state of material.

! f

i s the y i e l d condition where YT i s the following tensor: YT = E

E

T-1 / ^P

,

E = ( K

,

n = lN i s the family of the internal parameters.

nit. r v v l u t i o t i equation f o r the i n t r r n a l parametrrs i s in thc following form:

The velocity of anelastic deformation i s defined by:

1 -1

(7) g P = t v P } -

= E B _ ~ - g = + [ < i @ q f ~ ,

T ) > > + x . - s

- - ^{as I} E-'

all.

A

>

0 i f =O and

k

= 0 , and A = 0 i f £<O o r

k-<

0

,

and the symbol < R > denotes the positive p a r t of r e a l R, and

The function i s the viscoplastic p a r t of t h i s gradient and must be defined i n an experimental way.

l h v heat equttlion givesth(. rat(, of t ~ m ~ ) ( ~ r a t u r c ~ a s a function of thc, total drformation, of the plastic deformation, and

oZ

the state of the materials:

(9) cc

i

⁼^M_ +

*

_a _{K n}⁺^;,, ⁽

^pp

i n the case of adiabatic processes, where c, i s the specific heat at constant deformation and

-

¹

C. i s the stored internal energy, M_ =

-

!, A_

.

I n n r e c t a n g u l a r coordinate system X. the motion i s described by x= x(X,t). The gradients of elastic and plastic transformation asskme the following form:

The Cauchy tensor has only one component u i n the coordinate system X.. The velocity of elastic deformation can be w r i t t e n i n the form: 11

where v = ax/at

,

E i s Young's modulus, a i s the c o e f f i c i e n t of linear thermal expansion and P i s the densit;: p = pp, / F F~ The velocity of plastic deformation i s defined by:

P 11 2 2 '

(12) D ₁₁= E ₁₁

6

₁₁^P-l₁₁^{E - ~}₁₁ = Ell

< I $ (

^Yll, ^K

,

T ) + A

-

a Y with equation of evolution f o r the internal parameter 11

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(with only one parameter K - s t r a i n hardening), The functions ($I and 2' a r e determined onthe basis of experimental investigations.

1 he heal equation (9) i n the case of adiabatic processes takes the form:

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i = o ,, I;

^pl1-1 ^{/ p} ^-

^~i

₁₁^E₁₁

^E

₀^{/ P o} ^{- -}^{a &}_a~ ^I; _' The system of equations (1 1) - (14) together with the equation of motion (15) 2 1F~~ = p o 6 ₁

and relation

(16) Dll = DT1

+

Dp 11

enable a unique statement of the problem of adiabatic wave propagation i n the elastic-viscoplastic b a r under given boundary and i n i t i a l conditions.

We introduce the nominal s t r e s s : S = F~ 22 a

,

and the logarithmic mcsures of dcfor- mations:

(17) e = I n E l l

11

,

fll = I n F l l

,

pl, = I n Pll

,

f22 = I n F 22 ' The system of equations (1 1) - (16) takesthe following form:

;? = exp (f ) 1 5 - 2 s

i 2 2 + ~

^(ill ^{- 2 ;}

11 11 .

-

¹¹

^I

⁺^{2 i}

We can solve this system of equations with the following i n i t i a l conditions:

(19) ell(X,O) = fll(X,O) = f22(X,0) = p _{1 1}(X,O) = 0

,

^T(X,O) ⁼^TO

,

and the boundary conditions i n the form:

( 2 0 ) v ( O , t ) = w

,

f o r t < t

,

S(0,t) = 0 f o r t h t : and S ( l 0 , t ) = 0 f o r t > O , where t i s the time of rebound of the sample.

k

I I I

-

DESCR I PT I O N OF EXPER IMENTS

The test stand i s based on Hopkinson p r e s s u r e b a r system and a thermovision set

-

F i g . 1.

I t i s designed f o r dynamic compression of c y l i n d r i c a l samples and f o r thermovision recording of temperature d i s t r i b u t i o n

-

c f . ref.4. The Hopkinson p r e s s u r e b a r system consistsof a com- pressed a i r gun (1) and a set of b a r s (3,4). The tested sample (2) leaves the gun b a r r e l at a high velocity, s t r i k e s against the face of the b a r and becomes considerably deformed. The sample velocity i s mesured by means of a l a s e r (9), the beam of which i s s p l i t b y a set of prisms (10) into two p a r a l l e l beads directed toward the phototransistors ( 8 ) . As a r e s u l t of the impact, an elastic wave of compression i s produced i n the measuring b a r and after reflec- tion t r a v e l s back i n t h i s b a r as an elastic wave of extension. A potentiometric s t r a i n gauge system (7) i s mounted at the mid-length of the bar. I t s signal i s processed i n an amplifier (17) and recorded i n one of the channels of a digital storage oscilloscope (15). T h i s signal represents the i n i t i a l and reflected elastic waves as function of time.

The impact point of the sample at the instant of s t r i k i n g against the active measuring h a r i s observed by a thermovision camera (12). The camera i s located at a distance of 300 mm from the geometric a x i s of the b a r s and the gun tube so as t o take a sharp image i n the detector

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C5-116 JOURNAL DE PHYSIQUE

plane. T e m p e r a t u r e v a r i a t i o n i s determined b y a n a l y s i n g the a m p l i f i e d d e t e c t o r signal of the i n f r a r e d r a d i a t i o n emitted b y t h e sample s u r f a c e p o i n t s l y i n g along the l i n e p a r a l l e l to the a x i s of the sample, that i s , b y a n a l y s i n g the signal at the t h e r m o v i s i o n image l i n e . T h i s

F i g . 1 - Schematic d i a g r a m of experimenral stand f o r d y r a m i c c o m p r e s s i o n cf c y l i n d r i c a l sam- l e s w i t h t h e r m o v i s i o n t e m p e r a t u r e r e c o r d i n g .

F i g . 2 - T y p i c a l c u r v e of sample r a d i a t i o n p o w e r v e r s u s X

signal i s o b s e r v e d and r e c o r d e d on the s c r e e n of an o s c i l l o s c o p e attachement 18. (14) : H(t) i n Fig.2.

The sample w e r e made of a soft aluminium a l l o y P A 1 . T h e c y l i n d i c a l samples w e r e do =9.8mm i n d i a m e t e r and lo = 95mm i n length - F i g . 3 . The c a r r y i n g p a r t o f the sample was l i m i t e d b y t w o t e f l o n r i n g s i n o r d e r to e n s u r e c o a x i a l motion of the sample i n s i d e the gun b a r r e l . T h e sample h a d no s i d e contact i n the motion i n s i d e the gun ~ l l i ( 11 mod(. it po%sil,lr l o a v o i d f r . i r t i ( , ~ ~ e f f e c t s and m i n i s i n t e r p r e t a t i o n of the measurement r e s u l t s .

The p o w e r of the r a d i a t i o n o b s e r v e d b y the camera v a r i e s both i n the time and space. T h e p r i n - c i p e of o p e r a t i o n of the t h e r m o v i s i o n c a m e r a used i n the p r e s e n t experiments consists i n obser-

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vation of the successive point of a selected sample l i n e by means of a rotating prism. I n the diagram H(X 1, a complete observation cycle i s contained between two points at which the quantity observed changes i t s sign. I n the same diagram prepared f o r a s t r i k e r velocity of 1 w = 70m/s

-

i t i s possible to distinguish three maxima - cf.Fig.2. The f i r s t maximum ( r a t h e r weak) located at the left i s caused by the heat radiated frorr the indicator heater w i r e s (point R); the second maximum (most distinct at point C), and the t h i r d maximum (at point D) c o r r e s - pond t o the end of the s t r i k e r b a r and the end of the incident b a r , respectively, both ends being heated up d u r i n g impact.

Fig.3

-

Sample.

F r o m the r e c o r d of deformation E (t) i n the incident b a r i t i s possible to determine the s t r a i n and stress at the contact point of the sample and the b a r (under the assumption that the incident b a r i s elastic). Permanent longitudinal and transversal deformations A and A

ctivel y were measured (Fig.4 - sample a f t e r deforrration). 1 1 22 respe-

Fig.4

-

Sample a f t e r deformation.

I V

-

A N A L Y S I S OF EXPERIMENTAL RESULTS. COMPAR I SON WITH THEORETICAL CAL- CULAT l ONS

An initial-boundary value problem i s solved f o r the same boundary and i n i t i a l conditions as those used i n the experiments described i n the § I I I . Numerical data were close t o those used f o r the sample tested i n the experiments. The numerical calculations have beell made in ! w o different cases:

(a) - I n the r e l a t i o n f o r the velocity of anelastic deformation ( 7 ) the il~st;~l~t;~nc,ous plasticity tvr.ni i s ncylc,ctc.d: so \v(. c.olisid(.r. o ~ i l y visc.c~l~lics!ic. ti(.fo~.n~icliolis

-

s ( . ( - r.t~f. 1

.

(b) - We take into account i n (7) both members: instantaneous p l a s t i c i t y and viscoplasticity.

The calculed permanent longitudinal and transversal deformations, A and A i n both cases ( f o r a f i x e d lime) of the sample under test a r e shown i n ~ i g ' . k i n comparcson with 22.

the corresponding s t r a i n s measured on the deformed samples. The maximum deformations appear on the boundary of the sample under test. I t i s seen that the maxima calculated i n the case (a) and (b) and measured coincide. We observe that the deformations A and A2 calculated i n the case (b) has the best correspondence with experimental results. I t followsTromthe 1 experiments described h e r e that i n the theoretical discussion on long dynamically deformed samples made of strain-rate-sensitive material i t i s necessary to take into account not only

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JOURNAL DE PHYSIQUE

the deformations of a relaxation type,but also the instantaneous deformations.

Fig.5 - Permanent deformations of sample: A longitudinal deformations, AZ2 transverse deformations : - -

- -

experimental,

-

t h e o r e t ~ c a l ^{1 1} ^. (calculated).

Fig.6 -Temperature increase along sample under test, on line X =v ( t

-

td ).

-

--

experimental,

-

thedretical (calculated). ¹ ^cam

Dashed line i n Fig.6 represents temperature v a r i a t i o n AT along the sample (that is,along the straight l i n e X = v (t - t

)I

The d i s t r i b u t i o n of the power of i n f r a r e d radiation along the

1 .

sample i s shown In

h:?l .

~ o f i d lines i n Fig.6 represent the temperature increase (also along the l i n e X i = vcam( t - td)) calculated from the numerical solution o f the i n i t i a l -boundary value problem. The irrpact velocity of the sample was almost identical i n both cases. The difference between the calculated and measured r e s u l t s i s greater i n the case (a), the best approximation of the experimental c u r v e i s given by case (b). The experimental temperature

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increase and the heated region of the sample face a r e l a r g e r that those predicted by the nume- r i c a l calculations.

REFERENCES

/ 1 / Nowacki, W .K. and Kurcyk, T., J .Mbc.Thbor.et Appl

.

,Numbro spbcial (1 962) 109-1 24.

/2/ Mandel, J . , Plasticiti? classique et viscoplasticit&, C I S M Publications,Spr i n g e r V e r l a g (1971).

/3/ Mandel, J . , C.R.Acad.Sci.Paris,Skrie B,= (1980) 5-7.

/4/ Malatynski, M. and Nowacki, W.K. and Oliferuk, W., Arch.of Mech.

,%