Thermomechanical analysis of the Taylor impact test

Diego J. Celentano

Citation: Journal of Applied Physics 91, 3675 (2002); doi: 10.1063/1.1435836

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 JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 6 15 MARCH 2002

Thermomechanical analysis of the Taylor impact test

Diego J. Celentanoa)
Departamento de Ingenierı́a Mecánica, Universidad de Santiago de Chile, Av. Bdo. O’Higgins 3363,
Santiago, Chile
共Received 31 July 2001; accepted for publication 19 November 2001兲
A coupled thermomechanical analysis of the Taylor impact test considering flat-ended cylindrical
specimens of different materials fired at several striking velocities is presented. To this end, a large
strain rate-dependent thermoelasto–plasticity model is used to define the constitutive relationships
appearing in the momentum and energy equations both solved in the context of the finite element
method. The influence of the strain, temperature, and strain rate effects on the material response is
particularly analyzed. The numerical predictions are experimentally validated and compared with
results provided by previously reported simulations and simplified analytical expressions. © 2002
American Institute of Physics. 关DOI: 10.1063/1.1435836兴

I. INTRODUCTION II. THERMOMECHANICAL FORMULATION

The perpendicular impact of cylindrical specimens on a The governing local equations describing a general ther-
rigid flat target is an important experimental procedure, ge- momechancial process can be expressed by the mass conser-
nerically known as the Taylor impact test,1 for determining vation, the equation of motion, the energy balance, and the
dissipation inequality 共all of them valid in ⍀⫻ ␥ , where ⍀ is
the mechanical behavior of materials subjected to high strain
the spatial configuration of a body and ␥ denotes the time
rates. Several experimental, theoretical and numerical studies
interval of interest with t苸⌫兲 respectively written in a La-
have been devoted to this problem during the last decades
grangian description as:10
共see Ref. 1–9 and references therein兲. The complex nature of
the corresponding governing equations has led to the devel- ␳ J⫽ ␳ 0 , 共1兲
opment of robust simulation techniques with the purpose of “• ␴⫹ ␳ b f ⫽ ␳ ü, 共2兲
obtaining approximate descriptions of the phenomena in-
volved in the impact process. ⫺ ␳ cṪ⫺“•q⫹ ␳ r⫺T ␤:d⫹ ␳ r int⫽0, 共3兲
The aim of this work is to perform a thermomechanical
analysis of the Taylor impact test considering flat-ended cy- ⫺q•“T⫹D int⭓0, 共4兲
lindrical specimens of different materials fired at several together with appropriate boundary and initial conditions and
striking velocities in order to assess and discuss the influence adequate constitutive relations for the Cauchy stress tensor ␴
of the strain, temperature, and strain rate effects on the ma- 共which is symmetric for the nonpolar case adopted in this
terial response. To this end, the coupled thermomechanical work兲, the tangent specific heat capacity c, the heat flux vec-
formulation proposed here to simulate the deformation pro- tor q, the tangent conjugate of the thermal dilatation tensor
cess that takes place during the impact is described in Sec. II. ␤, the specific internal heat source rint , and the internal dis-
This large strain rate-dependent thermoelasto–plasticity- sipation Dint . In these equations, “ is the spatial gradient
based formulation includes the definitions of an isotropic- operator, the superposed dot indicates time derivative, and
specific free energy function and plastic evolution equations the subscript 0 applied to a variable denotes its value at the
that are the basis of all the material constitutive relations. initial configuration ⍀ 0 . Moreover, ␳ is the density, u is the
The corresponding finite element model is briefly presented displacement vector, J is the determinant of the deformation
in Sec. III. It should be mentioned that this finite element gradient tensor F 共F⫺1 ⫽1⫺“⫻u, with 1 being the unity
formulation is an alternative approach to existing methodolo- tensor兲, b f is the specific body force vector, T is the tempera-
gies dealing with large plastic deformations. ture, r is the specific heat source, and d is the rate-of-
deformation tensor (d⫽1/2(“⫻v⫹v⫻“), where v⫽u̇ is
The numerical simulation of the Taylor impact test ap-
the velocity vector兲. In classical thermodynamics, the mate-
plied to cylindrical specimens made of mild steel 共SAE
rial behavior during a thermomechanical process can be de-
1020兲, Armco™ iron, and oxygen-free high conductivity
scribed by means of a specific Helmholtz free energy func-
共OFHC兲 copper is detailed in Sec. IV. The numerical results
tion ␺ whose definition in terms of some state variables
obtained with the proposed formulation are experimentally
allows the derivation of ␴, c, ␤, r int and D int . 11 The free
validated and, additionally, compared with other available energy definition adopted in this work is valid for small elas-
simulations and results provided by simplified analytical ex- tic strains and isotropic material response 共both assumptions
pressions. being normally accepted for metals and other materials兲 con-
sidering the Almansi strain tensor e⫽1/2(1⫺F⫺T •F⫺1 ),
a兲
Electronic mail: dcelenta@lauca.usach.cl where T is the transpose symbol, the temperature, and a set

0021-8979/2002/91(6)/3675/12/$19.00 3675 © 2002 American Institute of Physics

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 3676 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

of nint phenomenological internal variables ␣k 共accounting evolution equations for such plastic variables are written as:
for the nonreversible effects and usually governed by rate
equations with k⫽1, . . . ,n int兲 as state variables of the L ␯ 共 ep 兲 ⫽␭˙ ⳵ F/ ⳵ ␶, eថ p ⫽⫺␭˙ ⳵ F/ ⳵ C p , ␨˙ p ⫽␭˙ ⳵ F/ ⳵ T, 共5兲
formulation.12 In this context, the Coleman’s method12 pro- where ␶ is the Kirchhoff stress tensor ( ␶⫽J ␴), L ␯ is the
vides the following relationships: ␴⫽ ␳ ⳵ ␺ / ⳵ e, ␩ ⫽⫺ ⳵ ␺ / ⳵ T well-known Lee 共frame-indifferent兲 derivative, ␭˙ is the plas-
2
is the specific entropy function, c⫽⫺T ⳵ ␺ / ⳵ T 2 , ␤ tic consistency parameter computed according to classical
2
⫽⫺ ␳ ⳵ ␺ / ⳵ e⳵ T⫽⫺ ⳵␴/ ⳵ T, r int⫽⫺1/␳ 0 (T ⳵ qk / ⳵ T⫺qk ) * D ␣k / concepts of the plasticity theory,14 C p is the total hardening
Dt and D int⫽qk * D ␣k /Dt, where qk ⫽⫺ ␳ 0 ⳵ ␺ / ⳵ ␣k are the function, and F⫽F( ␶,C p ) is the yield function governing
conjugate variables of ␣k and, according to the nature of the plastic behavior of the solid such that no plastic evolu-
each internal variable, the symbols* and D(•)/Dt appering tions occur when F⬍0. A Von Mises yield function is
in the previous expressions respectively indicate an appropri- adopted:
ate multiplication and a time derivative satisfying the princi-
pal of material frame-indifference.10 Furthermore, the heat F⫽ 冑3J 2 ⫺C p , 共6兲
flux vector at the spatial configuration is assumed to be given where J 2 is the second invariant of the deviatoric part of ␶
by the Fourier’s law written as q⫽⫺k“T, where k is the and C p is assumed to be given by:
conductivity coefficient. Additionally, a more restrictive dis-
sipative assumption than that stated in Eq. 共4兲 reads: ⫺q
p p
C p ⫽A p 共 ē 0p ⫹ē p 兲 n 具 1⫺T * m 典 关 1⫹B p ln共 eថ p /eថ ref
p
兴, 共7兲
•“T⭓0 and D int⭓0.10 The first condition is automatically
fulfilled for k⭓0 while the second imposes restrictions over where A p , B p , n p , and m p are hardening material constants,
the constitutive model definition. T * is the homologous temperature 关 T * ⫽(T⫺T ref)/(T m
The internal variables and their corresponding evolution ⫺T ref), with T m and T ref respectively denoting the melting
equations are defined in this work within the associate rate- and reference temperature兴, 具 典 are the Macauley brackets
dependent thermoplasticity theory context.13,14 A possible 关具 x 典 ⫽xH(x) where H is the Heaviside function: H(x)⫽1 if
choice is given by the plastic Almansi strain tensor ep, the x⬎0 and H(x)⫽0 otherwise兴, eថ ref p
is a reference value of the
effective plastic strain ē p related to the isotropic strain hard- effective plastic strain rate eថ , and ē 0p is an assumed initial
p
p
ening effect and the plastic yield entropy ␨ p. 12 The consider- value of ē p such that C th ⫽A p ē 0pn where C th is the static
ation of ␨ p in this thermomechanical context is consistent yield strength 共elastic limit兲 at the reference temperature. Eq.
with the principle of maximum plastic dissipation.15,16 The 共7兲, known as the Johnson–Cook relationship,4 clearly shows

FIG. 1. Impact of a mild steel specimen: computed characteristic parameters versus impact velocity for different initial lengths and diameters: 共a兲 Ratio
L f /L 0 , 共b兲 ratio R f /R 0 and 共c兲 impact duration.
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 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano 3677

TABLE I. Cases studied.

Initial length Initial diameter Impact velocity Nominal strain rate Initial temperature
Material L 0 关m兴 D 0 关m兴 U 关m/s兴 U/L 0 关1/s兴 T 0 关°C兴

Mild steel 122 4 800

共SAE 1020兲 0.025 4 0.007 11 183 7 200 25
244 9 600
0.007 87 338 13 300
Armco™ iron 0.025 4 0.007 61 197 7 750 25
221 8 700
OFHC copper 130 5 100
0.025 4 0.007 61 146 5 750 25
190 7 480
138 4 600 22
213 7 100
0.030 0 0.006 00 138 4 600 452
208 6 930
135 4 500 962
181 6 030

that C p grows with an increase in the effective plastic strain pic elastic constitutive tensor, cs is the secant specific heat
and the effective plastic strain rate, but drops with an in- capacity and eth is the thermal Almansi strain tensor given
crease in the temperature of the material 共thermal softening by:
effect兲. In this context, note that eថ p ⫽ 冑2/3dp :dp , with dp
⫽L ␯ (ep ) being the plastic rate-of-deformation tensor. eth ⫽ 21 关 1⫺ 共 1⫺a th 兲 2/3兴 1, 共9兲
With these considerations, the following specific free en-
ergy function ␺ ⫽ ␺ (e⫺ep,ē p , ␨ p,T) is proposed: where a th ⫽ ␣sth (T⫺T ref)⫺ ␣th0
s
(T 0 ⫺T ref) with ␣sth being the
secant volumetric thermal dilation coefficient. If a th Ⰶ1, note
1 1 that eth ⫽a th /31 which is, in fact, the classical expression for
␺⫽ 共 e⫺ep 兲 :Cs : 共 e⫺ep 兲 ⫺ 共 e⫺ep 兲 :Cs :eth the infinitesimal thermal strain tensor.
2␳0 ␳0
The partially coupled elastic/plastic decomposition of ␺
1 expressed by Eq. 共8兲 can be considered nowadays to be well
⫹c s 关共 T⫺T ref兲 ⫺T ln共 T/T ref兲兴 ⫹ 共 e⫺ep 兲 :Cs : ␶0 established 共see Ref. 13 and references therein兲, and it has
␳0
been used extensively in different isothermal and nonisother-
1 p mal applications.12–16 The stress–strain law 共secant or hyper-
⫺ ␩ 0 共 T⫺T 0 兲 ⫹ ␺ 0 ⫺ A p 共 ē 0p ⫹ē p 兲 n ⫹1
共 n ⫹1 兲 ␳ 0
p elastic form for the Cauchy stress tensor兲, the specific en-
tropy function, the tangent specific capacity, the tangent
p 1 p conjugate of the thermal dilatation tensor, the specific inter-
⫻ 具 1⫺T * m 典 ⫺ ␨ T, 共8兲
␳0 nal heat source, and the expression of the internal dissipa-
tion, all of them obtained from the above definition of ␺, are
where Cs is the secant 共measured with respect to T ref兲 isotro- respectively given by:

TABLE II. Properties of the mild steel 共SAE 1020兲, Armco™ iron and OFHC copper.

Mild steel Armco iron OFHC copper

Young modulus E 关MPa兴 210 000 182 000 117 000

Poisson ratio ␯ 0.3 0.26 0.35
Thermal dilation coefficient ␣sth 关°C⫺1兴 10⫺5 1.19⫻10⫺5 10⫺5
Static yield strength C th 关MPa兴 333 333 100
Hardening coefficient A p 关MPa兴 731.7 431.7 430.7
Hardening coefficient B p 0.050 0.060 0.025
Hardening coefficient n p 0.186 7 0.146 7 0.386 7
Hardening coefficient m p 1.0 1.0 2.7
Melting temperature T m 关°C兴 1 525 1 525 1 060
Reference temperature T ref 关°C兴 25 25 25
Reference strain rate ē˙ p 关s⫺1兴
ref
1.0 1.0 1.0
Initial density ␳ 0 关kg/m3兴 7 800 7 870 8 920
Secant specific heat capacity c s 关J/共kg °C兲兴 450 450 460
Conductivity k 关W/共m °C兲兴 52 52 380

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 3678 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

TABLE III. Impact of a mild steel specimen: final dimensional ratios. hardening contribution related to the local equilibrium state
defined by ␺ while the second one accounts for rate-
U
关m/s兴 L f /L 0 X/L 0 R f /R 0 dependent effects. According to Ref. 13, note that this addi-
tional term is included in the specific internal heat source and
Simulation 122 0.923 0.500 1.179
(B p ⫽0) 共0.898兲 共0.470兲 共1.261兲
internal dissipation expressions, respectively, given by Eqs.
共B p ⫽0 & m p ⫽⬁兲 共0.899兲 共0.470兲 共1.250兲 共14兲 and 共15兲. Finally, it is worth mentioning that the internal
Experimenta 0.940 0.450 ¯ dissipation inequality is verified owing to the adopted previ-
Simulation 183 0.855 0.400 1.368 ous definitions.
(B p ⫽0) 共0.809兲 共0.350兲 共1.536兲
共B p ⫽0 & m p ⫽⬁兲 共0.811兲 共0.350兲 共1.499兲
III. FINITE ELEMENT FORMULATION
Experimenta 0.870 0.390 ¯
Simulation 244 0.777 0.300 1.622 The spatial discretization of the coupled thermomechani-
(B p ⫽0) 共0.709兲 共0.280兲 共1.894兲
cal formulation presented above is carried out in the context
共B p ⫽0 & m p ⫽⬁兲 共0.714兲 共0.260兲 1.825
Experimenta 0.790 0.320 ¯ of the finite element method17–19 such that the global dis-
cretized thermomechanical equations 共equations of motion
a
See Refs. 1 and 2. and energy balance, respectively兲 both including mass con-
servation can be written in matrix form for a certain time t
as:12
1 RU ⬅FU ⫺MÜ⫺F␴ ⫽0,
␴⫽ 关 Cs : 共 e⫺ep ⫺eth 兲 ⫹ ␶0 兴 , 共10兲
J
RT ⬅FT ⫺CṪ⫺KT⫺Lint⫺GU̇⫽0, 共16兲
1
␩ ⫽⫺ 共 e⫺ep 兲 :Cs :e Tth 1⫹c s ln共 T/T ref兲 ⫹ ␩ 0 where RU and RT are the so-called mechanical and thermal
␳0
residual vectors, respectively. Moreover, FU is the external
p
1 p m p T * 共 m ⫺1 兲 force vector, M is the mass matrix, U is the nodal displace-
⫺ p A p 共 ē 0p ⫹ē p 兲 n ⫹1 ment vector, F␴ denotes the internal force vector, FT is the
共 n ⫹1 兲 ␳ 0 T m ⫺T ref
external heat flux vector, C is the capacity matrix, T is the
p 1 p nodal temperature vector, K is the conductivity matrix, Lint is
⫻H 共 1⫺T * m 兲 ⫹ ␨ , 共11兲 the internal heat source vector, and G is the thermoelastic
␳0
coupling matrix. The element expressions of these matrices
T p ⫹1 and vectors can be found in Ref. 12.
c⫽c s ⫺ A p 共 ē 0p ⫹ē p 兲 n
共 n p ⫹1 兲 ␳ 0 The integration of the terms containing time derivatives
p ⫺2 兲
of U and T in system 共16兲 is respectively carried out with the
m p 共 m p ⫺1 兲 T * 共 m p Hilber–Hughes–Taylor 共HHT兲 method and the generalized
⫻ H 共 1⫺T * m 兲 , 共12兲
共 T m ⫺T ref兲 2 midpoint rule algorithm choosing the corresponding param-
eters that make both procedures unconditionally stable.18 The
1 last scheme has also been used to integrate the plastic rate
␤⫽ Cs :e Tth 1, 共13兲
J equations 共5兲 via a return-mapping procedure.15,19
In order to overcome the volumetric locking effect on
1
r int⫽ 关共 TJ ␤⫹ ␴兲 :L ␯ 共 ep 兲 ⫹ 共 C p ⫺TC Tp 兲 eថ p 兴 , 共14兲 the numerical solution when incompressible plastic flows are
␳0 considered, like that resulting from the elasto–plastic model
described above, an enhanced strain-displacement matrix,
D int⫽ ␶ :L ␯ 共 ep 兲 ⫹C p eថ p ⫹T ␨˙ p , 共15兲
with e Tth ⫽ 31 (1⫺a th ) ⫺1/3␣sth and
p p ⫺1 兲
C Tp ⫽⫺A p 共 ē 0p ⫹ē p 兲 n 共 m p T * 共 m 兲 / 共 T m ⫺T ref兲
mp
⫻H 共 1⫺T * 兲关 1⫹B p ln共 eថ p /eថ ref
p
兲兴 .
For simplicity, the temperature derivative of the material
properties are neglected in these expressions. As discussed in
Ref. 12, the additive decomposition of the Almansi strain
tensor is recovered in this context through the multiplicative
decomposition of the deformation gradient into elastic, ther-
mal and plastic contributions. Moreover, the definition of ␺
given above also allows the computation of the conjugate of
the internal variables. It should be noted that the relations ␶
⫽⫺ ␳ 0 ⳵ ␺ / ⳵ ep and T⫽⫺ ␳ 0 ⳵ ␺ / ⳵␨ p are identically fulfilled.
However, the total hardening function given by Eq. 共7兲 is
FIG. 2. Impact of a mild steel specimen: computed total hardening function
obtained in this framework as C p ⫽⫺ ␳ 0 ⳵ ␺ / ⳵ ē p ⫹A p (ē 0p distribution along the rod axis at the end of the impact for different striking
p p
⫹ē p ) n 具 1⫺T * m 典 B p ln(eថ p/ēref
p
) where the first term is the velocities.
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 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano 3679

TABLE IV. Impact of a mild steel specimen: impact duration, average and maximum total hardening function,
p p
ratios C aver /C th and C max/Cth, maximum effective plastic strain and maximum temperature.

p p
U tf C aver C max T max
关m/s兴 关 10⫺6 s兴 关MPa兴 关MPa兴 p
C aver /C th p
C max/Cth p
ē max 关°C兴

Simulation 122 26.8 631.15 862.29 1.895 2.589 0.360 101.7

(B p ⫽0) 共35.9兲 共450.09兲 共621.43兲 共1.351兲 共1.866兲 共0.546兲 共105.6兲
共B p ⫽0 & m p ⫽⬁兲 共35.7兲 共446.32兲 共651.21兲 共1.340兲 共1.955兲 共0.520兲 共103.2兲
Derived from experimenta 22.0 870.07 ¯ 2.672 ¯ ¯ ¯
Simulation 183 33.0 685.24 932.48 2.057 2.800 0.770 205.6
(B p ⫽0) 共43.5兲 共483.82兲 共663.05兲 共1.453兲 共1.991兲 共1.264兲 共230.5兲
共B p ⫽0 & m p ⫽⬁兲 共42.9兲 共499.00兲 共750.61兲 共1.498兲 共2.254兲 共1.131兲 共218.2兲
Derived from experimenta 31.0 851.96 ¯ 2.617 ¯ ¯ ¯
Simulation 244 37.0 722.20 968.14 2.168 2.907 1.503 399.8
(B p ⫽0) 共48.1兲 共505.05兲 共669.99兲 共1.516兲 共2.011兲 共2.608兲 共475.4兲
共B p ⫽0 & m p ⫽⬁兲 共46.8兲 共529.05兲 共856.39兲 共1.588兲 共2.571兲 共2.280兲 共467.1兲
Derived from experimenta 37.0 871.86 ¯ 2.678 ¯ ¯ ¯
a
See Refs. 1 and 2.

classified within the B-bar methods and successfully checked been obtained with this method and with the classical New-
in many applications,12,16 is also used in the present analysis. mark algorithm which is, in fact, a particular case of the
This technique enables the use of simple elements and leads HHT scheme. Moreover, it is important to emphasize the
to a straightforward computational implementation. proper performance of the aforementioned improved isother-
The numerical solution of the resulting coupled finite mal split in the simulation of limiting situations 共e.g., prob-
element equations is achieved by using a staggered scheme15 lems with high impact velocity and/or high initial tempera-
in combination with an improved isothermal split, which pre- ture兲 in which the standard isothermal partition fails.
serves the coupling degree and provides unconditional Frictionless perfect contact is assumed at the impact
stability.12,16 In this framework, the well-known total La- zone. Since the heat transfer to the surrounding environment
grangian approach under isothermal conditions is considered during the impact can be considered negligible in this
for the mechanical problem, while the solution of the thermal problem,9 adiabatic conditions are adopted at the rod bound-
problem is done at the spatial configuration assuming a fixed aries. The transient thermomechanical response is computed
elastic configuration. It should be noted that this approach up to the time of deceleration t f at which the kinetic energy
differs from the standard conditionally stable isothermal par- is almost entirely dissipated 共the minimum time correspond-
tition in which the thermal problem is solved for a fixed ing to a zero velocity at the rear of the specimen has been
thermoelasto–plastic configuration. Furthermore, the taken as a criterion to obtain t f 兲. Furthermore, fracture and
linesearch technique is also employed during the Newton– damage effects as well as deformation twinning are not con-
Raphson iterative procedure to accelerate the numerical con- sidered in the present analysis.
vergence. The material responses of mild steel, Armco™ iron and
OFHC copper specimens during the Taylor impact test are
IV. NUMERICAL SIMULATION AND EXPERIMENTAL separately analyzed below.
VALIDATION
A. Mild steel
The finite element thermomechanical formulation briefly
The computed ratio of the cylinder final length L f to the
presented above is employed to simulate the behavior of flat-
initial one as a function of the striking velocity is presented
ended cylindrical specimens fired perpendicularly at a rigid
in Fig. 1共a兲 for different initial dimensions of the specimen.
target considering different materials, striking velocities, and
Similar results for the ratio of the final radius at the contact
initial temperatures. The cases studied in this work are sum-
face R f to the initial radius R 0 and the duration of impact are
marized in Table I. The thermomechanical material proper-
ties used in the analysis for mild steel 共SAE 1020兲, Armco™
iron and OFHC copper are presented in Table II.5,6
TABLE V. Impact of a mild steel specimen: average and maximum effective
In the numerical simulations, axisymmetric conditions plastic strain rate and average elasto–plastic front velocity.
are assumed with a uniform finite element mesh composed of
400 four-noded isoparametric elements 共10⫻40 elements in U p
ē˙ aver p
ē˙ max
p
U aver
the radial and axial directions, respectively兲 and a time step 关m/s兴 关1/s兴 关1/s兴 关m/s兴
of 0.1 ␮s that approximately corresponds, as normally sug- Simulation 122 5 475 33 000 3 400
gested for this kind of problem,9 to the time taken for a Derived from experimenta 4 370 ¯ 564
dilatation elastic wave to travel through an element in the Simulation 183 9 510 57 000 3 550
mesh. It should be mentioned that the algorithmic damping Derived from experimenta 5 900 ¯ 390
provided by the HHT method to remove the participation of Simulation 244 15 100 90 000 3 720
Derived from experimenta 7 060 ¯ 324
the hf modal components has not been absolutely necessary
a
in the present analysis, since very similar solutions have See Refs. 1 and 2.
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 3680 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

FIG. 3. Impact of a mild steel specimen: computed effective plastic strain evolution at points of the impact face for different striking velocities: 共a兲 122 m/s,
共b兲 183 m/s, and 共c兲 244 m/s.

respectively given in Figs. 1共b兲 and 1共c兲. As expected, lower while no significant changes in the material response are ob-
and higher values of L f /L 0 and R f /R 0 are correspondingly served when the thermal softening phenomenon is neglected.
obtained for increasing impact velocities.1–3 In the range of Figure 2 depicts the computed total hardening function
moderate striking velocities plotted, longer impact durations distribution along the rod axis at the end of the impact for
are also observed for larger values of U. The well-known fact three striking velocities. Higher values can be appreciated in
stating that the ratio L f /L 0 is practically independent of L 0 those zones with larger deformations, confirming once more
and D 0 is confirmed by these results. On the other hand, the the relative importance of the strain and strain rate effects
ratio R f /R 0 depends exclusively on R 0 . Moreover, the ob- over the thermal softening in this particular application. Note
tained linear variation of the impact duration with L 0 for a that plastic deformation takes place in the whole specimen
given striking velocity agrees with the approximate since the condition C p ⬎C th is attained even at the rear part
expression:1 t f /L 0 ⫽2(1⫺L f /L 0 )/U, which is also indepen- of the projectile. In agreement with the experimental
dent of D 0 .
observations,2,3 little change in C p within the range of strik-
Table III presents simulated results for the ratios L f /L 0 ,
ing velocities studied has been detected in the simulations.
X/L 0 共X being the low-strained length measured from the
The analytical expression obtained in Ref. 1 for the total
rear of the specimen computed, according to Ref. 2, as the
hardening function assumes a constant distribution of this
length of the specimen with final radius less than 3% of R 0 兲
variable along the rod. The values given by such an expres-
and R f /R 0 for three striking velocities. These final dimen-
sional ratios are also computed with simplified hardening sion using the experimental measurements of Ref. 2 are
laws in order to assess the effects of the strain rate and tem- shown in Table IV for different impact velocities. As can be
perature on the material response. Additionally, available ex- seen, these values are closer to the highest peaks of the com-
perimental results for L f /L 0 and X/L 0 are included for com- puted spatial distributions shown in Fig. 2 than to their av-
p
parison purposes.2 In general, the predictions corresponding erage values C aver . This can also be appreciated in the rea-
p
to the complete hardening law are those that better approach sonable fitting of C max /Cth obtained in the simulation with
the experimental data. Remarkably, the ratio L f /L 0 is prop- the dynamic yield strength–elastic limit ratio derived from
erly described by the simulation. However, some discrepan- the experiments. Furthermore, underestimated total harden-
cies between the experimental and numerical results are ing functions are obtained by neglecting strain rate and tem-
found for X/L 0 , presumably attributed to the inherent diffi- perature effects in Eq. 共7兲.
culty in the determination of this variable. The effect of the Computed and experimentally derived impact durations
strain rate on the final length is apparent in these results, are compared in Table IV showing a fairly good agreement.
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 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano 3681

FIG. 4. Impact of a mild steel specimen: computed effective plastic strain evolution at points of the rod axis for different striking velocities: 共a兲 122 m/s, 共b兲
183 m/s, and 共c兲 244 m/s.

The inclusion of strain rate effects in the hardening law is The effective plastic strain evolutions obtained with the
seen to be crucial in order to predict realistic values of t f . simulation at points located at the impact face for different
The maximum values of the effective plastic strain and striking velocities are shown in Fig. 3. In these three cases,
temperature are also summarized in Table IV. The influence the plastification starts at the edge and moves progressively
of the thermal softening and strain rate effects on the trends towards the center of the rod, where the maximum value is
exhibited by ē p and T can clearly be noted, i.e., the hardest reached at advanced times of the process. Another common
behavior occurs when both phenomena are considered, while feature that can be deduced from these curves is that the
the condition B p ⫽0 gives the softest material response. effective plastic strain evolutions at the impact zone clearly
stop before the end of the collision is attained. The fact that
larger values of ē p are obtained for increasing impact veloci-
ties is also an expected result ratified by the simulations.
Figure 4 presents computed effective plastic strain evo-
lutions at points of the rod axis. The final values of ē p gradu-
ally decrease for larger distances from the impact face up to
small, nearly constant distributions predicted at the rear part
of the rod, showing, as commented above, that the whole
specimen is plastically deformed. These curves also illustrate
that the strain rate effects are relevant mainly in regions near
the impact zone.

TABLE VI. Impact of a mild steel specimen: average force at the impact
end.

U ␳ L 0 ␲ R 20 U/t f 1/t f 兰 t f Fdt

关m/s兴 关kN兴 关kN兴

122 35.807 35.687

FIG. 5. Impact of a mild steel specimen: computed effective plastic strain 183 43.619 43.413
contours at the end of the impact for different striking velocities: 共a兲 122 244 51.872 51.332
m/s, 共b兲 183 m/s, and 共c兲 244 m/s.
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 3682 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

FIG. 6. Impact of a mild steel specimen: computed temperature evolution at points of the rod axis for different striking velocities: 共a兲 122 m/s, 共b兲 183 m/s,
and 共c兲 244 m/s.

The average and maximum effective plastic strain rates The computed effective plastic strain contours at the end
obtained from Fig. 4 are presented in Table V and compared of the impact are plotted in Fig. 5. Very high gradients can be
with the average value calculated through the simplified one- appreciated at the impact face. Similar trends have been
dimensional expression1 U/2(L 0 ⫺X) using the experimental found in Refs. 3– 6 and 9 for different materials and impact
measure of X.2 The uniform deceleration assumed in such velocities.
analytical expression leads to lower values than those corre- The temperature evolutions predicted by the simulation
sponding to the nominal strain rates 共see Table I兲. However, at points of the rod axis can be seen in Fig. 6 for three impact
an opposite trend is exhibited by the computed average strain velocities. Great temperature variations caused by plastic de-
rates which, in addition, are found to vary linearly with U. It formation are observed in these cases. In particular, note the
should be noted that the nonuniform strain rate distributions direct relationship between the evolutions of ē p 共see Fig. 4兲
present peak values of nearly six times greater than the av- and T at these points, reflecting the fulfillment of adiabatic
erage ones. conditions during the impact. Moreover, the development of
large temperature gradients at the end of the collision are due

FIG. 8. Impact of a mild steel specimen: experimental 共dots and solid line兲
FIG. 7. Impact of a mild steel specimen: computed elasto–plastic front and predicted 共deformed mesh兲 final shapes for a striking velocity of 338
position evolution along the rod axis for different striking velocities. m/s.
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 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano 3683

FIG. 10. Impact of an OFHC copper specimen: results at the end of the
impact for a striking velocity of 190 m/s: 共a兲 Computed effective plastic
FIG. 9. Impact of an Armco iron specimen: results at the end of the impact strain contours and 共b兲 experimental 共dots and solid line兲 and predicted
for a striking velocity of 221 m/s: 共a兲 computed effective plastic strain 共deformed mesh兲 final shapes.
contours and 共b兲 experimental 共dots and solid line兲 and predicted 共deformed
mesh兲 final shapes.

⫽3 217 m/s, which corresponds to the velocity of a one-

to the nonuniform effective plastic strain distribution 共see dimensional elastic shear wave that also involves no dilata-
Fig. 5兲. The oscillations present in these evolutions are gen- tion. On the other hand, the elastic front related to the dila-
erated by elastic volumetric contractions and dilatations oc- tation wave is estimated by tracking the evolution of the
curring in the course of the deformation process. associate stress invariant I 1 ⫽tr( ␶) 共where tr is the trace sym-
Figure 7 depicts the computed elasto–plastic front posi- bol兲 along the specimen length. This elastic front location is
tion evolutions along the rod axis for different striking ve- found to be practically independent of the striking velocity,
locities. These evolutions, derived from the data plotted in with an average front velocity of approximately 6 400 m/s,
e
Fig. 4, are nearly independent of U and involve no dilation which agrees relatively well with the analytical value20 U dil
according to the elasto–plastic model considered in the ⫽(E(1⫺ ␯ )/ ␳ 0 (1⫺2 ␯ )(1⫹ ␯ )) ⫽6020 m/s which corre-
1/2

simulation. As can be seen, the elasto–plastic front reaches sponds to the velocity of a one-dimensional elastic irrota-
the end of the rod at early times of the impact process. The tional wave of dilatation. The distance travelled by this wave
computed average elasto–plastic front velocities shown in is almost the original rod length, owing to the short time
Table V are higher than those calculated analytically1 con- elapsed until the end of the specimen is reached 共about 3.8
sidering experimental measurements of L f and X, since such ␮s兲.
one-dimensional approximation assumes that the plastically For the purpose of assessing the performance of the time
distorted portion of the projectile is given by the reduced integration scheme, the average force developed at the target
length L f ⫺X, in sharp contrast with the results provided by plate during the impact obtained in the simulation using the
the present simulation in which, as already mentioned, the Newmark method is contrasted with the simple relationship
development of plastic effects occur in the whole rod length. expressed as the initial linear momentum divided by the im-
As L f ⫺X is found to be independent of U 共see Table III兲, the pact duration. The results for different striking velocities are
decreasing trend of the experimentally derived values of presented in Table VI. The excellent agreement achieved be-
p
U aver with U is explained by the increase of t f with higher tween these two forms of computing the average reaction
impact velocities 共see Table IV兲. The large values of U aver
p
shows the correct conservative behavior of the Newmark al-
predicted by the simulation are possibly due to the relevant gorithm in this application.
three-dimensional effect on the stress tensor evolution that An experimentally available final shape of a cylindrical
takes place during the impact. Moreover, it shoud be noted mild steel fired at a striking velocity of 338 m/s 共see Table I兲
that the computed average elasto–plastic front velocities ap- is contrasted with the corresponding simulation results. The
proach the analytical value20 U shear e
⫽(E/ ␳ 0 2(1⫹ ␯ )) 1/2 typical mushroom shape, characterized by a large radius in-

TABLE VII. Impact of a mild steel specimen: final dimensions, impact duration, maximum total hardening
function, maximum effective plastic strain and maximum temperature.

p
U tf C max T max
关m/s兴 L f /L 0 X/L 0 R f /R 0 关 10⫺6 s兴 关MPa兴 p
ē max 关°C兴

Simulation 338 0.651 0.200 2.222 41.0 989.11 3.040 851.0

Experimenta 0.660 0.220 2.220 42.0 ¯ ¯ ¯
a
See Refs. 1 and 2.
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 3684 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

TABLE VIII. Impact of an Armco™ iron specimen: final dimensions, impact duration, maximum total hard-
ening function, maximum effective plastic strain and maximum temperature.

p
U tf C max T max
⫺6
关m/s兴 L f /L 0 R f /R 0 关 10 s兴 关MPa兴 p
ē max 关°C兴

Simulation: present work 197 0.807 1.647 42.9 666.76 1.715 324.4
Simulation: Johnson–Cook modela 0.795 1.650 ¯ ¯ ¯ ¯
Simulation: dislocation-based modela 0.816 1.660 ¯ ¯ 1.600 337.0
Experimenta 0.802 1.590 ¯ ¯ ¯ ¯
Simulation: present work 221 0.768 1.781 45.0 663.11 2.278 422.0
Simulation: Johnson–Cook modela 0.755 1.800 ¯ ¯ ¯ ¯
Simulation: Johnson–Cook modelb 0.779 1.802 ¯ ¯ 1.800 ¯
Simulation: dislocation-based modela 0.779 1.810 ¯ ¯ 2.000 407.0
Simulation: dislocation-based modelb 0.779 1.810 ¯ ¯ 1.910 ¯
Experimenta 0.780 1.800 ¯ ¯ ¯ ¯
a
See Ref. 5.
b
See Ref. 6.

crease at the impact face with a nearly constant radius at the worth mentioning the adequate performance of the proposed
rear region of the rod, can be observed with dots and a solid numerical formulation in spite of the huge element distor-
line in Fig. 8. The deformed mesh is also plotted for com- tions that develop during the impact. Moreover, computed
parison. As can be seen, the numerical predictions are in and experimental final dimensions and impact duration, to-
good agreement with the experimental measurements. The gether with some predicted maximum characteristic values,
differences in shape existing in the lateral edge between both are all summarized in Table VII. Once again, these results
results are due to the fact that the depth of the dent in the experimentally validate the proposed thermomechanical for-
target plate was not negligible in the experiments.2 It is mulation.

TABLE IX. Impact of a OFHC copper specimen: final dimensions, impact duration, maximum total hardening function, maximum effective plastic strain and
maximum temperature.

p
U T0 tf C max T max
关m/s兴 关°C兴 L f /L 0 R f /R 0 关 10⫺6 s兴 关MPa兴 p
ē max 关°C兴

Simulation: present work 130 0.787 1.418 62.4 468.88 0.881 100.8
Simulation: Johnson–Cook modela 0.812 1.480 ¯ ¯ ¯ ¯
Simulation: dislocation-based modela 0.784 1.410 ¯ ¯ 0.000 167.0
Simulation: one-dimensional modelb 0.819 ¯ ¯ 462.0 ¯ ¯
Experimenta 0.770 1.300 ¯ ¯ ¯ ¯
Simulation: present work 146 0.753 1.507 63.7 500.11 1.110 127.7
Simulation: Johnson–Cook modela 0.778 1.580 ¯ ¯ ¯ ¯
Simulation: dislocation-based modela 25 0.750 1.490 ¯ ¯ 1.100 167.0
Simulation: one-dimensional modelb 0.772 ¯ ¯ 507.33 ¯ ¯
Experimenta 0.736 1.400 ¯ ¯ ¯ ¯
Simulation: present work 190 0.661 1.786 65.4 572.16 2.100 257.6
Simulation: Johnson–Cook modela 0.682 1.920 ¯ ¯ ¯ ¯
Simulation: Johnson–Cook modelc 0.638 1.803 ¯ ¯ 2.040 ¯
Simulation: dislocation-based modela 0.658 1.770 ¯ ¯ 1.800 297.0
Simulation: dislocation-based modelc 0.638 1.776 ¯ ¯ 1.570 ¯
Simulation: one-dimensional modelb 0.708 ¯ ¯ 546.72 ¯ ¯
Experimenta 0.638 1.780 ¯ ¯ ¯ ¯
Simulation 138 0.763 1.442 75.3 492.88 0.944 107.1
Experimentd 22 0.755 ¯ ¯ ¯ ¯ ¯
Simulation 213 0.599 1.927 77.2 640.99 2.606 321.5
Experimentd 0.600 ¯ ¯ ¯ ¯ ¯
Simulation 138 0.706 1.588 91.3 396.52 1.424 548.8
Experimentd 452 0.688 ¯ ¯ ¯ ¯ ¯
Simulation 208 0.532 2.206 90.7 533.84 3.872 730.5
Experimentd 0.530 ¯ ¯ ¯ ¯ ¯
Simulation 135 0.516 2.445 148.0 147.46 4.322 1050.6
Experimentd 962 0.526 ¯ ¯ ¯ ¯ ¯
Simulation 181 0.369 3.248 135.9 219.83 5.282 1054.0
Experimentd 0.354 ¯ ¯ ¯ ¯ ¯
a
See Ref. 5.
b
See Ref. 7.
c
See Ref. 6.
d
See Ref. 8.
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 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano 3685

B. Armco™ iron clearly different from that found for the Armco™ iron 共see
Fig. 9兲. The reason is due to their different hardening param-
Computed final dimensions, impact duration, and maxi-
eters B p and n p 共see Table II兲, i.e., lower B p coefficients
mum characteristic values corresponding to the present simu-
promote larger R f and shorter L f , while higher n p param-
lation for two striking velocities are shown in Table VIII.
eters induce shorter R f and L f . For a similar impact velocity
Available experimental5 and previously reported numerical
results5,6 are also included. In general, a very good agree- 共e.g., ⬃190 m/s兲, the ratio L f /L 0 is smaller in the OFHC
copper than that in the Armco™ iron, while an opposite
ment between the experimental and numerical results can be
trend is obtained for R f /R 0 , demonstrating, for this particu-
observed. Some small discrepancies existing in the different
lar case, the relative importance of ē p over the strain rate
simulations that use the Johnson–Cook hardening law given
effects in the hardening law for the OFHC copper. This fact
by Eq. 共7兲 may be attributable to the various options adopted
is also apparent in the short length X encountered for the
in such calculations for the stress–strain relationship, the
deformed specimen shown in Fig. 10.
time integration scheme, and the numerical treatment of the
The computed results also confirm that the deceleration
plastic incompressibility constraint. Furthermore, note that
is mainly due to material strength, i.e., the greater the
these results are close to those predicted by the dislocation-
strength, the faster the deceleration for a given impact veloc-
based hardening model5,6 where, in particular, a remarkably
ity independently of the material.3 This can be observed in
good fitting is achieved for the maximum values of ē p and T.
Tables IV, VIII, and IX for a striking velocity of about 190
Figure 9 shows the computed effective plastic strain con-
m/s.
tours at the end of the impact, together with the experimental
Similar results to those previously discussed are also
and simulated final shapes of the projectile for a striking
presented in Table IX for three different initial temperatures
velocity of 221 m/s. It is seen that the final profile is satis-
共22, 452, and 962 °C兲, considering two striking velocities for
factorily predicted. In addition, these numerical results agree
such cases. The experimental validation of the present for-
well with the computed contours obtained in Refs. 5 and 6.
mulation is restricted to the available experimental measures
of L f /L 0 . 8 A very good quantification of this ratio is pro-
C. OFHC copper
vided by the simulation throughout the studied T 0 and U
Table IX shows final dimensions, impact duration and ranges. These results allow the extension of the validity of
maximum characteristic values obtained with the present for- the C p ⫺t f relationship discussed earlier to any initial tem-
mulation for three impact velocities at an initial temperature perature. Moreover, the L f /L 0 ⫺U curves for each initial
of 25 °C. Some already published experimental5 and numeri- temperature are plotted in Fig. 11. The ratio R f /R 0 and the
cal results5–7 are also presented to validate the proposed ap- effective plastic strain grow with T 0 for a given U. Within a
proach. Although the one-dimensional model7 seems to be moderate range of striking velocities, as discussed above, the
slightly inaccurate at high impact velocities, an overall good impact duration directly increases with U for low values of
agreement can be observed between all these results. The the initial temperature. However, this trend changes at high
maximum values of C p , ē p , and T predicted in this work initial temperatures. The fact that the maximum total harden-
reasonably adjust the respective ones calculated with other ing function increases with U for a given T 0 reveals that the
models.5–7 strain and strain rate effects prevail on the thermal softening
The computed effective plastic strain contours at the end phenomenon even at high temperatures. Finally, it is seen
of the impact as well as the experimental and predicted final that the maximum temperature, which is also localized at the
shapes of the rod are all depicted in Fig. 10 for a striking center of the impact face for these cases, almost reaches the
velocity of 190 m/s. Once more, these numerical results are melting point for T 0 ⫽962 °C.
in accordance with those computed in Refs. 5 and 6. More-
over, it is seen that the deformed mesh fits the measured
profile. The deformation pattern for the OFHC copper is
V. CONCLUSIONS

A coupled thermomechanical analysis of the Taylor im-

pact test has been presented. To this end, a finite element
large strain thermoelasto–plasticity-based formulation has
been proposed and used to simulate the material response
during the impact. This study was focused on cylindrical
flat-ended specimens made of mild steel, Armco™ iron and
OFHC copper fired at several velocities. The influence of the
strain, temperature, and strain rate effects on the material
behavior has been also assessed by using the well-known
Johnson–Cook hardening relationship. The numerical pre-
dictions obtained in this work have been satisfactorily vali-
dated with available experimental data and compared with
previously reported simulations and results provided by sim-
FIG. 11. Impact of an OFHC copper specimen: ratio L f /L 0 versus impact plified analytical expressions. In particular, the impact dura-
velocity for different initial temperatures. tion and final dimensions of the specimen have shown an
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 3686 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Diego J. Celentano

overall good agreement between the experimental and nu- ACKNOWLEDGMENTS

merical results in the range of impact velocities and initial
The supports provided by the Chilean Council of Re-
temperatures considered in the analysis. Although little ther-
search and Technology CONICYT 共FONDECYT Project No.
mal softening has been observed for low initial temperatures,
1990588兲 and the Department of Technological and Scien-
the importance of the strain rate effects on the material re-
tific Research at the University of Santiago de Chile
sponse has been confirmed by the simulations. Moreover, a
共DICYT-USACH兲 are gratefully acknowledged.
nonuniform total hardening function distribution has been
predicted along the rod according to the spatially variable 1
G. Taylor, Proc. R. Soc. London 194, 289 共1948兲.
evolutions of the strain, temperature, and strain rate for 2
A. Whiffin, Proc. R. Soc. London 194, 300 共1948兲.
which, specifically, large gradients have been detected near 3
M. Wilkins and W. Guinan, J. Appl. Phys. 44, 1200 共1973兲.
4
the impact zone. Furthermore, a nearly constant elasto– G. Johnson and W. Cook, in Proceedings of Seventh International Sym-
plastic front velocity has been obtained for different striking posium on Ballistics, The Hague, The Netherlands, 1983, p. 541.
5
F. Zerilli and W. Armstrong, J. Appl. Phys. 61, 1816 共1987兲.
velocities. This front was found to reach the end of the speci- 6
G. Johnson and T. Holmquist, J. Appl. Phys. 64, 3901 共1988兲.
men at early times of the collision promoting, therefore, the 7
S. Jones and P. Gillis, J. Appl. Phys. 61, 409 共1986兲.
development of plastic deformation in the whole rod length. 8
W. Gust, J. Appl. Phys. 53, 3566 共1982兲.
9
The computed large values of the elasto–plastic front veloc- R. Batra and J. Stevens, Comput. Methods Appl. Mech. Eng. 151, 325
共1998兲.
ity are attributable to the significant three-dimensional effect 10
L. Malvern, Introduction to the Mechanics of a Continuous Medium
on the stress tensor evolution that took place during the im- 共Prentice-Hall, Englewood Cliffs, NJ, 1969兲.
pact. The simulation also confirmed that the deceleration is 11
B. Coleman and M. Gurtin, J. Chem. Phys. 47, 597 共1967兲.
mainly due to material strength, showing that the greater the
12
D. Celentano, Int. J. Plast. 17, 1623 共2001兲.
13
J. Lubliner, Plasticity Theory 共Macmillan, New York, 1990兲.
strength, the faster the deceleration for a given impact veloc- 14
J. Simo, in Handbook of Numerical Analysis 共Elsevier, Amsterdam, 1995兲,
ity, independent of the material and initial temperature. Vol. III.
Moreover, the different deformation patterns found for the 15
F. Armero and J. Simo, Int. J. Plast. 9, 149 共1993兲.
16
D. Celentano, D. Gunasegaram and T. Nguyen, Int. J. Solids Struct. 36,
materials considered in the present analysis denoted the rela-
2341 共1999兲.
tive importance of the strain and strain rate effects in their 17
O. Zienkiewicz and R. Taylor, The finite element method, Fourth Ed., Vols.
respective hardening laws. Finally, it is worth mentioning 1 and 2 共McGraw-Hill, London, 1989兲.
18
that the improved isothermal split used in the simulation has T. Hughes, The finite element method. Linear static and dynamic analysis
共Prentice-Hall, Englewood Cliffs, NJ, 1987兲.
allowed, in contrast with the standard isothermal partition, 19
M. Crisfield, Non-linear finite element analysis of solids and structures
the analysis of limiting situations such as those involving 共Wiley, Chichester, 1991兲, Vols. 1 and 2.
large impact velocities and high initial temperatures. 20
H. Kolsky, Stress waves in solids 共Dover, New York, 1963兲.

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