Modeling dynamic crack propagation in fiber reinforced composites including frictional effects
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Mechanics of Materials 35 (2003) 481–509
www.elsevier.com/locate/mechmat
Modeling dynamic crack propagation in fiber
reinforced composites including frictional effects q
S.K. Dwivedi 1, H.D. Espinosa *
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA
Received 17 September 2001; received in revised form 29 April 2002
Abstract
Dynamic crack propagation in a unidirectional carbon/epoxy composite is studied through finite element analyses of
asymmetric impact (shear loading) of a rod against a rectangular plate. A finite deformation anisotropic visco-plastic
model is used to describe the constitutive response of the composite. Crack propagation is simulated by embedding zero
thickness interface element along the crack path. An irreversible mixed-mode cohesive law is used to describe the
evolution of interface tractions as a function of displacement jumps. Contact and friction behind the crack tip are
accounted for in the simulations. The failure of the first interface element at the pre-notch tip models onset of crack
extension. Crack propagation is modeled through consecutive failure of interface elements. The dynamic crack prop-
agation phenomenon is studied in terms of crack initiation time, crack speed, mode I and mode II displacement jumps
and tractions associated with the failure of interface elements, effective plastic strain at the crack tip and path inde-
pendent integral J 0 . Analyses are carried out at impact velocities of 5, 10, 20, 30 and 40 m/s, assuming the crack wake is
frictionless. Moreover, analyses at impact velocities of 30 and 40 m/s are also carried out with a friction coefficient of
0.5, 1, 5 and 10 along the crack surfaces. The analyses show that steady-state intersonic crack propagation in fiber
reinforced composite materials occurs when the impact velocity exceeds a given threshold. A steady-state crack speed of
3.9 times the shear wave speed and 83% of the longitudinal wave speed is predicted in the cases in which the impact
velocity is above 10 m/s. Detailed discussion is given on the features of sub-sonic and intersonic crack propagation. It is
shown that friction effects, behind the crack tip, do not have a significant effect on maximum crack speed; however, they
do on characteristics of the shock wave trailing the crack tip. The analyses also show that the contour integral J 0 ,
computed at contours near the crack tip, is indeed path independent and can serve as a parameter for characterizing
intersonic crack propagation.
Ó 2002 Elsevier Science Ltd. All rights reserved.
q
This article is based on presentations given at the ASME International Congress and Exposition, November 1998, Los Angeles,
CA, and an ARO-MURI review meeting. The research here reported was then published in the second authorÕs web page but otherwise
remained unpublished. The reader will find the status of the subject, reported analyses and conclusions as given in the original
presentations. Since then, several contributions have been made by various research groups. For completeness these contributions are
cited and discussed in the context of the reported numerical simulations.
*
Corresponding author. Tel.: +1-847-467-5989; fax: +1-847-491-3915.
E-mail address: espinosa@northwestern.edu (H.D. Espinosa).
URL: http://clifton.mech.northwestern.edu/espinosa.
1
Currently at Institute of Shock Physics, Washington State University, Pullman, WA 99164-2816, USA.
0167-6636/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 6 7 - 6 6 3 6 ( 0 2 ) 0 0 2 6 6 - 1482 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
1. Introduction Kumar and Kishore (1998). In these experiments,
as shown in Fig. 1, half of the bimaterial specimen
Dynamic crack propagation in materials has is made of a metal, aluminum or steel, and the
remained an active area of research for decades. In other half is made of a more compliant material,
the past few years, the importance of this area has e.g., PMMA or Homalite. The specimen is asym-
re-emerged, primarily due to (a) greater emphasis metrically impact loaded and the crack propagates
on multi-layered materials in strategic applica- dynamically under the influence of intense stress
tions, (b) role of dynamic crack propagation in waves. A number of important results have been
determining the overall response and damage reported from these experiments. At an impact
mode of advanced materials, and (c) experimental velocity of less than 5 m/s, the extension and dy-
findings showing that the velocity of a dynamically namic growth of the crack has been found to occur
growing crack can exceed the shear wave speed of predominantly in mode I. The maximum crack
the materials. It is this feature that has drawn speed, Vc , is observed to be 80% of the Rayleigh
much attention in recent years. wave speed, CR , of the more compliant material,
There have been a number of experimental i.e., PMMA or Homalite. These crack speeds
measurements on initiation and dynamic propa- Vc < Cs , where Cs is the shear wave speed of the
gation of cracks along bimaterial interfaces, Tippur more compliant material, have been termed sub-
and Rosakis (1991), Tippur et al. (1991), Rosakis sonic crack speeds. At an impact velocity of more
et al. (1991), Lambros and Rosakis (1995a,b,c), than 20 m/s, crack extension and propagation takes
Singh and Shukla (1996), Singh et al. (1997a), and place in predominantly shear or mode II. For these
Fig. 1. Schematic of dynamic crack propagation experiment and intersonic crack propagation event reproduced from Singh et al.
(1997a).S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 483
pffiffiffi
shear driven cracks, maximum speed of propaga- Cs < Vc < 2Cs , in which the crack growth is un-
tion has been found to be as high as 1.5Cs . These stable and the crack quickly accelerates out of this
crack speeds Cs < Vc < Cl , where Cl is the dila- region. In contrast, Singh et al. (1997a) have
tional wave speed of the more compliant material, shown that stable crack growth indeed takes place
have been termed intersonic crack speeds. in this regime. Instead of strain energy release rate
The intersonic crack propagation phenomenon, reducing monotonously to zero as in Lambros and
along a bimaterial interface, was at first puzzling. Rosakis (1995a), Kavaturu and Shukla (1998)
In fact, theory predicts that sustained mode I crack have shown that it first increases with crack speed
propagation in homogeneous materials, at a speed and then, after attaining a maximum, it starts de-
equal to the Rayleigh speed of the material, re- creasing and approaches zero.
quires an infinite amount of energy. Other inter- There are two obvious reasons for such dis-
esting observations were: (i) crack tip acceleration crepancies. First, experimentation with a smaller
in the intersonic region is reported to be as high as width of plates, Tippur and Rosakis (1991), Tip-
107 g, where g is the gravitational acceleration; (ii) pur et al. (1991), Rosakis et al. (1991), and
development of a contact zone and the formation Lambros and Rosakis (1995a,b,c), did not allow
of lines of discontinuity (shock waves) behind in- intersonic crack propagation to reach the steady
tersonically propagating cracks. state due to boundary effects. Second, the crack tip
Analyses of these experimental records have stress field solutions used to derive the above
been carried out using analytical solutions of the conclusions are based on the assumption that
two dimensional, singular, elastic crack tip stress stress fields are elastic, crack front extends infi-
fields problem, Rice et al. (1990), Wu (1991), Liu nitely and crack faces are traction-free. In actual
et al. (1993), Liu et al. (1995), Ricci et al. (1997), experiments, some of these assumptions are not
Singh et al. (1997b), and Kavaturu and Shukla met. Due to stress concentration, the crack tip is
(1998). It has been shown that the ratio between always associated with an inelastic regime. The
displacement along the crack and a direction nor- interaction between reflected stress waves and dy-
mal to it, remains constant behind the dynamically namic crack fronts is unavoidable. Furthermore,
propagating crack. This ratio is less than unity (of because of contact behind the crack the interson-
the order of 0.3) in the sub-sonic regime exhibiting ically propagating crack wake is not free of trac-
mode I cracking, whereas it is more than unity (of tion. Additionally, there are frictional effects in the
the order of 3.0) in the intersonic regime exhibiting contact zone which modify the crack tip stress
mode II cracking. Furthermore, the strain energy fields and its propagation characteristics. These
release rate decreases with the increase in crack observations illustrate the complexity of the problem
speed and approaches zero in the intersonic region. and the need for detailed numerical studies.
The analyses have also confirmed the existence of a There have been a number of computational
contact zone and formation of a Mach zone behind studies on bimaterial interface cracks, Shih and
intersonically propagating cracks. Asaro (1990), Varias et al. (1990a, 1990b), Yang
In spite of the these crucial findings, there are et al. (1991), Nakamura et al. (1995), Xu and
some discrepancies in the experimental character- Needleman (1995), Xu and Needleman (1996),
ization of dynamic crack propagation. Experi- Siegmund et al. (1997), Kumar and Kishore
mental studies by Tippur and Rosakis (1991), (1998), and Needleman and Rosakis (1999), to
Tippur et al. (1991), Rosakis et al. (1991), and mention a few. However, few analyses have been
Lambros and Rosakis (1995a,b,c), did not estab- conducted to study the intersonic crack propaga-
lish a steady-state intersonic crack propagation. tion and the effect of various factors. Nakamura
Using a larger width of bimaterial specimen, Singh et al. (1995) used the strain energy release rate as
et al. (1997a) showed that a steady-state intersonic the governing criteria for dynamic propagation of
crack speed can be achieved. Similarly, according cracks and found that the dynamic J 0 integral
to Lambros and Rosakis (1995a) and Liu et al. approaches zero with the increase in crack veloc-
(1995), there exists an unfavorable velocity regime, ity. Xu and Needleman (1996), and Needleman484 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 and Rosakis (1999), carried out an extensive para- gation under impact loading. In particular, the metric study with the help of assumed material examination of conditions for intersonic crack properties and obtained intersonic crack speeds propagation need to be assessed. Likewise, the fol- and a contact zone behind the propagating crack. lowing features need further research: (1) thresh- However, the interface cohesive elements used by old impact energy above which intersonic crack Xu and Needleman (1996), and Needleman and propagation is possible in unidirectional fiber Rosakis (1999), did not include frictional effects composites, (2) effects of plastic zone around the after failure and creation of free surfaces. It should crack tip, (3) formation of shock waves, (4) char- be noted that in their model the dynamic J 0 inte- acteristics of contact zone behind crack tip, (5) gral does not decrease with increasing crack speed. effect of friction in the contact zone, and (6) Recently, Rosakis et al. (1999) showed that in- characterization of intersonic crack propagation in tersonic shear cracks also occur in homogeneous terms of the dynamic J 0 integral. materials in which a weak plane pre-determines the To our knowledge, only a limited number of path of the crack. They found that these shear analytical and computational studies, on the sub- cracks pffiffiffi accelerate towards a particular speed of ject of dynamic crack propagation in fiber com- 2Cs , under steady-state conditions, as predicted posite materials, were reported in the literature. by theory. Likewise, a line of discontinuity or Among these studies we can mention Huang et al. shock wave develops at an angle of 45° with re- (1999), Hwang and Geubelle (2000), Kumar and spect to the plane of the crack. These findings il- Kishore (1998), Stout et al. (1998), Lo et al. (1993), lustrate that intersonic crack propagation is Sun and Qian (1997), Pandey and Sun (1996), and primarily related to the propagation mode and Sun and Wu (1996). None of these studies included that the phenomenon occurs as long as enough frictional contact behind the crack tip. The present energy is delivered to the crack tip. work focuses on studying dynamic crack propa- In the case of unidirectional fiber composites, a gation in unidirectional composites using a cohe- weak fracture plane typically develops at the inter- sive surface formulation and a contact/friction face between matrix and fibers. Furthermore, due to algorithm. Dynamic crack propagation in a unidi- material elastic anisotropy, wave speeds along the rectional graphite/epoxy composite plate is studied fiber direction are very different to the wave speeds using the finite deformation anisotropic visco- in the perpendicular direction. Hence, it was ex- plastic model, in total Lagrangian co-ordinates, of pected that the intersonic crack phenomenon could Espinosa et al. (2001). The contact/interface also occur in these materials as well. Experimental methodology of Espinosa et al. (1998b) is used to studies on dynamic crack propagation in unidirec- analyze crack propagation. The model is first tional fiber composites were conducted by Zheng validated with the analysis results of Xu and and Sun (1995), Lambros and Rosakis (1997), Stout Needleman (1996) for dynamic crack propagation et al. (1998), and Coker and Rosakis (1998, 2001). along a bimaterial interface. It is then used to The work by Coker and Rosakis revealed that study dynamic crack propagation in unidirectional asymmetrically loaded (mode II) cracks indeed graphite/epoxy composite plates under the influ- propagate intersonically at unprecedented speeds of ence of varying impact velocities and surface fric- three times the composite shear wave speed and tion characteristics. The dynamic J 0 integral is almost the material longitudinal wave speed along evaluated in each case to study the energy associ- the fiber direction. Evidence of large scale frictional ated with the propagating crack tip. contact was also reported. The same findings were computationally predicted by Espinosa et al. (1998a,b), see first page footnote. 2. Finite deformation anisotropic visco-plasticity Due to the importance of fiber composite ma- model terials in civilian and strategic defense applica- tions, it is important to develop tools for detailed The simulation of the response of fiber com- finite element analyses of dynamic crack propa- posite materials requires the formulation of large
S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 485
deformation inelastic constitutive models. Identi-
fication of an appropriate yield function and the
plastic flow rule to describe the non-linear aniso-
tropic behavior of fiber composite materials is a
challenge in itself, see e.g., Sun and Chen (1989),
Espinosa et al. (1997), Chen et al. (1997), OÕDon-
oghue et al. (1992), Voyiadjis and Thiagarajan
(1995) and Voyiadjis and Thiagarajan (1996), etc.
amongst others. Previously, the approach has been
to computationally homogenize the directional
properties of composite materials. This helps in
deriving a formulation applicable for such mate-
rials and in facilitating the implementation of a
finite element computer program. In contrast,
micromechanical models, Aboudi (1989), Paley
and Aboudi (1992) and Stout et al. (1998) treat the
matrix and fiber materials separately, and derive Fig. 2. Total Lagrangian continuum model.
the overall constitutive relation of a cell using pe-
riodicity and traction continuity at the boundary Z Z
of sub-cells. On one hand, homogenized constitu- P0 : r0 g dB0 q0 ðb0 aÞ g dB0
tive relations are simple to use in a finite element B0 B0
Z
code; however, their capability to simulate exten-
sive damage during impact and penetration re- t g dS0 ¼ 0 ð2Þ
S0r
mains a topic of research. On the other hand,
micromechanical cell models can simulate a large where P0 is the first Piola–Kirchhoff stress tensor
extent of damage modes, but they are highly at time t; b0 , a, and t are the body force vector,
computer intensive. Modeling all the inherent acceleration vector, and boundary traction vector
characteristics of fiber composite materials to de- on volume B0 and boundary S0r , respectively.
termine their response at high strain rates remains Virtual displacement field g is assumed to be ad-
a formidable challenge. Simplified models have missible, and q0 represents the material density per
been sought to study selected behavior of these unit volume in the reference configuration. The
materials, see for instance Lee and Sun (1992) and symbol r0 denotes the material gradient with re-
Lee and Sun (1993). In this work, one of such spect to the reference configuration, and Ô:Õ is used
simple models is employed. to denote the inner product between second order
Consider a solid with volume B0 in the reference tensors, e.g., A : B Aij Bji , where the summation
configuration, and a deformation process charac- convention on repeated indices is implied.
terized by the mapping xðX; tÞ (see Fig. 2). A Alternately, the weak form of the momentum
material point initially at X will be located at balance, in terms of spatial quantities, is given by
Z Z
x ¼ X þ u after deformation, in which u is the
displacement vector. A displacement based finite s : rs g dB0 q0 ðb0 aÞ g dB0
B0 B0
element formulation is obtained from the weak Z
form of the momentum balance or dynamic prin- t g dS0 ¼ 0 ð3Þ
S0r
ciple of virtual work. The weak form at time t in
total Lagrangian co-ordinates, i.e., referred to the in which superscript s stands for the symmetric
reference configuration, is given by part of the tensor, s ¼ FP0 is the Kirchhoff stress,
Z F is the deformation gradient at time t, and r is
½r0 P0 þ q0 ðb0 aÞ g dB0 ¼ 0 ð1Þ the spatial deformation tensor. As Eq. (3) shows,
B0 the equation of motion in its weak form states that486 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
the work done by the stresses s over strains rs g E_ ije ¼ E_ ij E_ ijp ð6Þ
equals the work done by applied body forces, in-
ertia forces, and surface tractions. In the absence In the above equation, E_ ijp is the plastic strain rate
of the body force, the above equation reduces to which is based on the associative flow rule,
the following discrete equation, MU € ¼ ðf int þ
of
f surf Þ, where M is the global mass matrix, U € is the E_ ijp ¼ k_ ð7Þ
oSij
global acceleration vector, f int is the internal force
vector and f surf is the equivalent surface force Here, f is the flow potential and k_ is the plastic
vector resulting from integration of the surface rate proportionality factor. The inelastic behavior
traction vector. The impactor as well as the target of the composite is modeled based on a flow po-
are discretized spatially into six node quadratic tential quadratic in the second Piola–KirchhoffÕs
triangular elements. The mass of triangular ele- stress tensor, as proposed by Espinosa et al.
ments is lumped at their nodes to obtain the global (1997), i.e.,
lumped mass matrix M. Details on the numerical 2 2 2
2f ðSij Þ ¼ a11 S11 þ a22 S22 þ a33 S33 þ 2a12 S11 S22
integration of the above equations can be found in
2
Espinosa et al. (1998b). þ 2a13 S11 S33 þ 2a23 S33 S22 þ 2a44 S23
The constitutive response of the fiber composite 2
þ 2a55 S13 2
þ 2a66 S12 ð8Þ
is formulated in terms of the Green–Lagrange
strain tensor E and the work conjugate second For unidirectional fiber reinforced carbon/epoxy
Piola–Kirchhoff stress tensor S, Espinosa et al. composite with 2–3 isotropy, 1 direction being the
(2001). The Green–Lagrange strain rate tensor at fiber direction and 2 normal to it in the lamina
time step t þ Dt, is given as, plane, the above yield function reduces to, Weeks
and Sun (1996) and Weeks and Sun (1998),
1
E_ ij ¼ ½ðFki Fjk ÞtþDt ðFki Fjk Þt ð4Þ 2
f ðSij Þ ¼ 12S22 2
þ a66 S12 ð9Þ
2Dt
where Fij is the deformation gradient and Dt is the Defining an effective stress as,
incremental time step. All quantities in the above pffiffiffiffiffiffi
S ¼ 3f ð10Þ
equation are defined in the global co-ordinates.
The second Piola–KirchhoffÕs stress rate tensor is and using Eqs. (8) and (10), the rate of plastic
given by a hyperelastic constitutive law, which in work is given by,
rate form is given by, _ p
W_ p ¼ Sij E_ ijp ¼ SE ð11Þ
S_ ij ¼ Cijkl E_ kl
e
ð5Þ
The proportionality factor of Eq. (7) is obtained as
where Cijkl is the elastic anisotropic material stiff- _ p
ness constitutive tensor in the global co-ordinates 3E
k_ ¼ ð12Þ
and E_ ije is the elastic component of the Green– 2S
Lagrange strain rate tensor. As explained by Es- where E _ p is the effective plastic strain rate. The
pinosa et al. (1998b), the Kirchhoff stress s and effect of strain rate and temperature can be mod-
second Piola–Kirchhoff stress S are related eled by defining the material strength in terms of
through the relation s ¼ FSFT . In the case of an- an effective stress which includes temperature and
isotropic materials, the elastic constitutive matrix rate terms, namely,
ðCl Þ is defined in the local co-ordinate system of the " #m
lamina. It is transformed to C ¼ TT Cl T to obtain _ ¼ E
p _ p
S
E 0 p; T Þ if S > gðE
p; T Þ ð13Þ
the constitutive matrix in the global co-ordinates gðE
following standard transformation procedures.
The elastic components of the strain rate tensor a
are obtained by an additive decomposition of the p T T0
gðE ; T Þ ¼ Sy 1 ð14Þ
total Green–Lagrange strain rate, namely, Tm T0S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 487
in which Sy is the flow stress at reference plastic be identified from off-axis and out-of-plane shear
strain rate E _ p and temperature T . T is the glass
0 0 m tests as explained in Espinosa et al. (2001).
transition temperature of the matrix material. S is
the flow stress at the current effective plastic strain
rate E_ p and temperature T . m and a are the rate
3. Contact/interface cohesive law
and temperature sensitivity exponents, respec-
tively. The reference flow stress Sy is defined by an The cohesive surface formulation has gained
experimentally found power law, prominence in simulating crack initiation, propa-
!1=n gation, and fragmentation in brittle materials, Xu
p
E and Needleman (1995), Xu and Needleman (1996),
Sy ¼ ð15Þ
A Xu (1996), Camacho and Ortiz (1996) and Espin-
osa et al. (1998a). Geubelle and Baylor (1998),
in which n is the strain hardening exponent. A Espinosa et al. (1999) and Espinosa et al. (2001)
summary of the constitutive equations in discrete have shown that a combined model incorporating
form is given in Table 1. Other functional forms homogenized constitutive relation for fiber com-
for g can be used, for instance, functions involving posite materials and cohesive interface elements
hyperbolic sines that are typical of polymer ma- between lamina can be employed to analyze dy-
trices. namic delamination in fiber composite materials in
To account for a nonlinear response during agreement with experiments.
unloading, another power law is used. The coeffi- In this study, the interface law used to simulate
cients used in the yield potential, given above, can onset and crack propagation is based on a contact/
interface approach proposed by Espinosa et al.
Table 1 (1998b) and Espinosa et al. (1999). As shown in
Summary of constitutive equations Fig. 3, interface elements are embedded along the
possible crack path. The tensile and shear tractions
S_ tþDt ¼ C E_ tþDt
e
developed by the interface element are calculated
from the interface cohesive law. This law describes
E_ tþDt ¼ E_ tþDt
e p
þ E_ tþDt the evolution of these tractions and the cou-
pling between them as a function of relative
of _ p of
3E normal and tangential displacements. The com-
p
E_ tþDt ¼ k_ ¼ _ p N
¼E
oStþDt 2S oStþDt tþDt tþDt pressive traction, if any, is calculated through the
2 3
of 0 2a66 S12 2a55 S13
3 oStþDt 3 4
NtþDt ¼ ¼ 2a66 S12 0 2a44 S23 5
2 StþDt 2S 2a S
55 13 2a44 S23 S33 tþDt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
StþDt ¼ 3ftþDt ðSij ÞtþDt
a
p T T0
tþDt
gðE ; T Þ ¼ Sy;tþDt 1
Tm T0
!1n
p
tþDt
E
Sy;tþDt ¼
A
" #m
_ p ¼ E_ p StþDt
E tþDt 0 p
tþDt
gðE ;TÞ
Fig. 3. Interface/contact model with finite kinematics.488 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
impenetrability condition employed in the contact The interface law II, Espinosa et al. (1999), is
module. Once the effective displacement jump ex- used to define the evolution of normal and tan-
ceeds a specified critical value, the interface ele- gential traction and the coupling between them as
ment is assumed to have failed, i.e., onset of crack a function of the relative normal and tangential
propagation is said to have taken place. The new displacement jumps, respectively. The equivalent
free surfaces as a result of crack propagation are traction sðkÞ is given in terms of a non-dimen-
automatically created. Thereafter, the interaction sionalized effective displacement jump parameter k
between the surfaces is described solely by the defined as:
contact algorithm, which includes frictional effects. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
The failure of successive interface elements simu- 2 2
Un Ut
lates crack extension as in Xu and Needleman k¼ þ ð16Þ
dn dt
(1996).
A detailed formulation of the interface element
where Un and Ut are the actual normal and tan-
and cohesive laws can be found in Espinosa et al.
gential displacement jumps at the interface. dn and
(1999), and Espinosa and Zavattieri (2002a,b).
dt are the critical normal and tangential displace-
Instead of a zero thickness 4-noded quadrilateral
ment jumps at which the interface fails or crack
element used by Espinosa et al. (1999), a zero
initiates in pure normal or pure shear mode, re-
thickness 6-noded interface element, as shown in
spectively. The value of k ¼ 0 signifies the un-
Fig. 4, is used in the present work. When inserted
stressed state of the element and k ¼ 1 signifies
at the boundary of 6-noded triangular elements
total failure. As described earlier, the tensile and
used in the discretization of the plates, the re-
shear state of the interface element for k 6 1 is
sponse of the interface element becomes more
determined from the interface cohesive law while
consistent to the discrete body response sur-
the compressive state is determined from the con-
rounding it. The shape functions of the one-
tact law. After complete debonding, i.e., k ¼ 1, the
dimensional 3-noded differential element are given
interface is characterized solely by the contact law.
in a standard textbook on finite element methods.
The procedure adopted in the explicit integration
Integration of surface traction to determine nodal
is summarized in Table 2.
force vector is carried out using a 3-point Gauss
It is assumed here that the interface traction
quadrature rule.
evolves reversibly up to a value k ¼ kc , similar to
the state of elastic loading and unloading in ma-
terials. Beyond kc , unloading from a state ku and
subsequent re-loading take place irreversibly. The
cohesive law used in the analyses is plotted in Fig.
5. The value of kc is selected such that the wave
speeds in the material with interfaces is the same as
the ones in the material without interfaces during
reversible loading. The interface law, so derived, is
given as,
Table 2
Contact and interface calculations based on interface traction
Load State Tn Tt
Tension-shear kS.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 489
Fig. 5. Cohesive law: evolution of interface traction in loading and unloading.
(i) for loading and unloading in the range In the absence of Mode II deformation, the area
0 6 k 6 kc : under the sn dn curve gives the Mode I critical str-
Un smax Ut smax ain energy release rate GIc , which can be written as:
sn ¼ ; st ¼ a ; ð17Þ
dn kc dt kc GIc ¼ 12dn smax ð21Þ
where sn and st are the normal and tangential Similarly, in the absence of Mode I deformation,
tractions corresponding to the normal and the area under the st dt curve gives the Mode II
tangential displacement jump Un and Ut re- critical strain energy release rate GIIc , which can be
spectively, smax is the maximum normal trac- written as:
tion beyond which failure initiates and a is a GIIc ¼ 12adt smax ð22Þ
mode mixity parameter;
(ii) for loading in the range kc < k 6 1: Hence, if dn ¼ dt , a defines the ratio GIIc =GIc (Es-
pinosa et al., 1999).
Unc 1 k Utc 1 k As mentioned earlier, failure of successive in-
sn ¼ smax ; st ¼ asmax
dn 1 kc dt 1 kc terface elements simulates crack propagation. For
ð18Þ this purpose, the time of failure of each Gauss point
along interface elements is recorded. This data
where Unc and Utc are the normal and tan-
provides the crack length history. The crack speed
gential displacement jumps attained at k ¼ kc ,
is obtained by differentiating the crack length data
(iii) for unloading and reloading in the range
using a four-point backward difference method. As
0 6 k 6 ku , where ku > kc is the last value of k
expected, the crack speed so obtained will have
from where irreversible unloading took place:
oscillations depending on the element size.
Un smax Ut smax
sn ¼ ; st ¼ a ð19Þ
dn ku dt ku
4. Dynamic J 0 integral
(iv) for loading in the range ku < k 6 1:
Unu 1 k The dynamic J 0 integral formulation was devel-
sn ¼ smax ; oped to establish a single parameter to characterize
dn 1 ku
Uu 1 k the dynamic stress field and energy associated with
st ¼ asmax t ð20Þ a dynamically growing crack, Nakamura et al.
dt 1 ku
(1995), Xu and Needleman (1996) and Kumar and
where Unu and Utu are the normal and tan- Kishore (1998). It is not the purpose here to review
gential displacement jumps attained at k ¼ ku the development of the formulation of J 0 and its
during reverse loading. numerous versions for different loading conditions,490 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
such as elasto-static, elasto-dynamic, elasto-plastic, All integrals are carried out over the area enclosed
and elasto-plastic-dynamic loading of the crack tip. by the outer contour C0 . As Pji oUi =oXj is the en-
In the present study, for a local co-ordinate system ergy term, the contribution by the second integral
xi where x1 is aligned parallel to the direction of is neglected in the present analyses. Various
crack propagation, the dynamic J 0 integral is eval- schemes for selecting the weighting function q has
uated from a closed contour around the crack tip as been used in the past (Carpenter et al., 1986). In
discussed in Atluri (1982), Nakamura et al. (1985), the present analyses, q is unity from the crack tip
Carpenter et al. (1986), Moran and Shih (1987) and up to the node just before the outer contour, then
Nishioka (1994), reduces linearly to zero at the outer contour. Also,
Z the component of q normal to the crack front and
0 oUi
J ¼ lim ðW þ T Þn1 Pji nj dC ð23Þ interface elements are taken as zero.
C !0 C oX1
The dynamic J 0 integral is evaluated for a num-
Z ber of contours using contour integrals as well as the
0 oUi
J ¼ ðW þ T Þn1 Pji nj dC weight function area integral. As per conventional
C0 oX 1
Z 2 practice, the parts CCþ and CC are assumed parallel
o Ui oUi oUi o2 Ui to each other and traction-free. Due to reasons of
þ q 2 q dA ð24Þ
A0 ot oX1 ot oX1 ot possible large rotation of the crack front and con-
where Pij are the ij components of the first Piola– tact zone behind the crack tip, these crack fronts are
Kirchhoff stress tensor, q is the material density, W not neglected. In order to account for these two
and T are the stress–work density and the kinetic crack faces, contours are formed through the nodes
energy density per unit initial volume defined as: of 6-noded quadrilateral elements, instead of
Z t Xt forming through the integration points (Li and Shih
W ¼ S : E dt ¼ Sij DEij ð25Þ (1985)). For calculating J 0 based on the contour
0 0 integral, the stress–work density W and Pij , obtained
2 from Sij , are extrapolated to contour nodes from the
1 oUi contributing integration points. However, the cal-
T ¼ q ð26Þ
2 ot culation of J 0 from the weight function area integral
and ni is the normal to contour C. C is the van- involves integration over the elementÕs area. Hence,
ishing inner contour around the crack tip, taken as no extrapolation is required in this case and calcu-
the crack tip itself in the present analyses. C0 is the lation is performed using standard finite element
outer contour including the crack faces, i.e., procedures. As shown in the following sections, the
C0 ¼ Cþ J 0 calculated from the contour integral equation
C þ C þ CC . A0 is the area between the in-
ner and outer contours. Since the inner contour is (24) and area integral equation (27) are identical.
taken at the crack tip, A0 becomes the area en-
closed by the outer contour.
Using the weight function q, the dynamic J 0 5. Analyses
integral is calculated as:
Z The above model is first validated by compari-
0 oUi oqk oqk son with analyses reported in the literature. Then it
J ¼ Pji ðW þ T Þ dA0
A0 oX1 oXj oXk is used to study dynamic crack propagation in
unidirectional carbon fiber reinforced composite
Z
o2 U i oW materials.
þ Pji qk dA0
A0 oXj oXk oXk
5.1. Model validation by comparison with literature
Z results
o2 Ui oUi oUi o2 Ui
þ q qk : q qk dA0
A0 ot2 oX1 ot oX1 ot Our integrated model is used to analyze dy-
ð27Þ namic crack propagation along bimaterial inter-S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 491 faces and the results are compared with the liter- figure for comparison. It is seen that the present ature data (Xu and Needleman, 1996). The prob- model is able to predict crack extension time and lem being analyzed is shown in Fig. 6 on the left crack speed in good agreement with the literature (Fig. 1a in Xu and Needleman (1996)). A bima- data, even though the latter used a very fine mesh terial plate of width 6.1 mm and height 5.6 mm with an interface element length of 0.004 mm. The with a central crack of length 0.5 mm is restrained far field path independent integrals calculated at at the bottom and loaded with a specified constant the 5th contour for the two cases are shown in Fig. velocity at the top. Two cases are considered. In 9 and compared with the literature results. It is one case, referred to as PMMA/MA the top half seen that the J 0 value for MA/PMMA agrees well plate is made of PMMA and bottom half plate is with the literature data. For the PMMA/MA case, made of an assumed material called A. The spec- there is qualitative agreement, but the J 0 value ified constant velocity in this case is V ¼ 30 m/s. In obtained in the present work differs from the lit- the other case, referred to as MA/PMMA, the two erature data. The observed differences in the crack material halves are opposite to the first case and speed or J 0 may be due to the coarser mesh used in the applied velocity is V ¼ 11:73 m/s. Due to the present analyses, differences in interface cohe- symmetry, only half of the plate is analyzed in each sive law as used in the analysis, difference in de- case. The displacement and velocity boundary fining the onset of crack extension, as well as to the conditions used in the two elastic analyses are more refined contact/interface methodology used shown in Fig. 6 in the center. In both cases, the in the present analysis. In the present investiga- velocity rises linearly from time zero to its maxi- tion, the crack is said to begin extension as soon as mum value at 0.1 ls as shown in Fig. 6 on the right an interface element fails, i.e., the displacement side. The finite element mesh with two zoomed jump at all the three integration points reaches the views at the pre-notch tip and the five contours critical displacement jump, i.e., k ¼ 1. Xu and considered for the J 0 calculation are shown in Fig. Needleman (1996) define the crack tip position 7. The mesh used for the analyses has 6712 nodes, where the displacement jump is five times the 3214 triangular elements and 89 interface elements critical displacement jump. Contrary to the present with average length of interface elements as 0.03 model, their approach ignores contact behind the mm. The properties of the two materials and in- crack tip and the initial slope of the cohesive law is terface, as used in the analyses, are given in Table quite different. The compressive part of their in- 3. terface cohesive law allows some interpenetration The crack speed obtained from the simulations at the interface, ahead as well as behind the crack are shown in Fig. 8. The analyses results of Xu and tip. The contact/interface approach of the present Needleman (1996) are also plotted in the same model enforces impenetrability strictly and hence Fig. 6. Problem geometry: model validation with analyses data for dynamic crack propagation along a bimaterial interface.
492 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
Fig. 7. Finite element mesh and contours used in the analyses.
is bound to modify the crack tip stress field.
Table 3 Nevertheless, the agreement obtained even with
Materials properties for validation with literature analyses the coarse mesh used in the present analyses is
Materials properties Units reasonable.
PMMA Material A
5.2. Unidirectional fiber composite plate loaded in
Density
q0 ¼ 1190:0 q0 ¼ 5950:0 kg/m3
mode II
Elastic properties The above analyses show that the proposed
E ¼ 3:24 E ¼ 6:44 GPa
m ¼ 0:35 m ¼ 0:15
model predicts the phenomenon of dynamic crack
Cs ¼ 1004 Cs ¼ 686 m/s propagation in close agreement with analyses re-
CR ¼ 938 CR ¼ 616 m/s sults taken from literature. Detailed analyses are
Interface properties
carried out next to study the phenomenon of dy-
Tmax ¼ 162 MPa namic crack propagation in an unidirectional
dn ¼ 2:1753 lm graphite/epoxy composite. The problem definition
dt ¼ 0:933 lm is shown in Fig. 10. A pre-notched unidirectional
a ¼ 2:3315 composite plate of total width 250 mm and height
kc ¼ 0:05
300 mm is considered. The fiber direction is takenS.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 493
mm. In order to minimize numerical oscillation, it
is assumed that the velocity increases to the maxi-
mum over a time period of 1.0 ls and remains
constant thereafter at the specified value of V m/s.
The problem geometry and boundary conditions so
defined are consistent with the experimental set up
reported in Coker and Rosakis (2001). In our cal-
culations the plate dimensions are slightly larger;
however, this difference does not modify the
physics of the problem and, as it will be discussed
later, most of the features observed experimentally
are captured in our simulations.
Since all materials and interface properties are
not available for one particular graphite/epoxy
composite, they are judiciously selected from two
sources. Elastic and anisotropic flow potential
data for the unidirectional composite plate are
Fig. 8. Crack speed predicted by present analyses and com- taken from Weeks and Sun (1996), while the in-
parison with literature data.
terface properties are taken from Stout et al.
(1998). It is assumed that this combination of
mechanical and interface properties provides an
idealized unidirectional carbon/epoxy composite
for the purpose of studying the phenomenon of
dynamic crack propagation. The material proper-
ties used in the analyses are summarized in Table
4. Note that these elastic constants are not the
same to the ones reported in Coker and Rosakis
(2001), in fact the work here reported was done
simultaneously and independently. Hence, the nu-
merical analyses here reported show that inter-
sonic crack propagation is a phenomenon that
may occur in a variety of unidirectional fiber
composites.
The top and bottom half of the composite plate
are discretized into 6-noded quadratic triangular
elements, while 6-noded quadratic interface ele-
ments are embedded between the two halves. As
Fig. 9. Plot of dynamic integral, J 0 , history and comparison shown on the left in Fig. 10, crack extension ini-
with literature data. tiates at the pre-notch tip and is allowed to prop-
agate up to a total width of 200 mm. The last 50
parallel to the X-direction. The length and maxi- mm width of the plate is considered bonded
mum opening of the pre-notch, parallel to the X- without any interface elements to avoid interface
axis, are 25 and 1.5 mm, respectively. The plate is failure from the impact face. The finite element
impact loaded off-center on the face opposite to the mesh size is a minimum around the crack path and
pre-notch face. The loading is provided by speci- gradually increased towards the top/bottom
fying a constant velocity in the negative X-direction and right side of the plate. The mesh has 52,032
over an assumed projectile diameter. The projectile nodes, 25,602 triangular elements and 295 inter-
diameter in all the analyses is kept constant as 50 face elements. The average size of the quadratic494 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
Fig. 10. Dynamic crack propagation in unidirectional fiber reinforced composite––problem definition (left) and specified impact ve-
locity (right).
Table 4 views and one enlarged view at the pre-notch tip
Materials properties of unidirectional carbon/epoxy composite are shown in Fig. 11 together with the five con-
Materials properties Units tours used to evaluate the dynamic path indepen-
Properties Symbol Value dent integral J 0 . Starting from the nearest contour
Density q0 1580 kg/m3 to the crack tip, contours are numbered serially
outwards as shown in the bottom plot of Fig. 11.
Elastic properties C11 131.83 GPa
C22 ¼ C33 13.90 GPa Analyses are carried out at five different impact
C44 ¼ C55 6.00 GPa velocities (V) of 5, 10, 20, 30 and 40 m/s and for
C66 3.45 GPa different coefficients of friction along the crack face
C12 ¼ C21 6.68 GPa behind the propagating tip. The crack faces are
C13 ¼ C31 6.68 GPa
assumed frictionless for the impact velocities of 5,
C23 ¼ C32 6.98 GPa
10 and 20 m/s. Whereas, the cases of V ¼ 30 and
Stress wave Cl 9134 m/s 40 m/s are analyzed for both the frictionless case as
speeds Cs 1949 m/s
well as with friction coefficients of 0.5, 1, 5 and 10.
CR 1811 m/s
It was found that propagating crack interacts with
Yield function a66 2.0 – the stress field generated at the beginning of the
Plastic flow S0 1.00 Pa bonded interface (last 50 mm of the plate width).
properties 0p 6 0:76%:
For E This interaction modifies the crack tip stress field
A ¼ 1:26 10 44 Pa and the features of crack propagation. For this
n ¼ 5:1 –
0p > 0:76%: reason, analyses results are discussed for a maxi-
For E
A ¼ 1:76 10 72 Pa mum crack extension of 150 mm, i.e., while the
n ¼ 8:5 – interaction is not present. Table 5 summarizes the
_ p 1
various performed analyses including impact ve-
Flow rate E 0 1:0 s
properties 10 5 locity and friction coefficient. A discussion of the
m 40.0 – results is given in the following sections.
Interface cohesive Tmax 85.00 MPa
law parameters dn 7.06 lm 5.2.1. Crack length and crack speed
dt 17.70 lm Fig. 12 shows the evolution of crack length (left)
a 1.20 – and crack speed (right) obtained from analyses
kc 0.01 – corresponding to a frictionless crack wake. Crack
extension initiates at 58.4, 51.6, 36.1, 32.4 and 31.2
interface elements, along the allowed crack path, is ls, when the composite plate is impacted at veloc-
0.1 mm. The full plate mesh with two zoomed ities of 5, 10, 20, 30 and 40 m/s, respectively. TheS.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 495
Fig. 11. Finite element mesh and contours for J 0 calculation used in the analyses.
crack initial speed for the five impact velocities are
209, 382, 658, 1829 and 2146 m/s. After initiation, Table 5
Summary of analyses for fiber composite plate
the crack tip accelerates and attains steady-state
Analysis Impact velocity Friction coefficient
speed before stopping or before interacting with
(m/s)
boundary waves. The crack acceleration is found to
1 5.0 0.0
be of the order of 107 g for impact velocities of 5
2 10.0 0.0
and 10 m/s and of the order of 109 g for impact 3 20.0 0.0
velocities of 20, 30 and 40 m/s, where g is the 4 30.0 0.0, 0.5, 1.0
gravitational acceleration. The maximum crack 5 30.0 5.0, 10.0
speed at the five impact velocities is obtained as 6 40.0 0.0, 0.5, 1.0
7 40.0 5.0, 10.0
1856, 1879, 7512, 7600, and 7600 m/s.496 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
Fig. 12. Evolution of crack length (left) and crack speed (right) at different impact velocities and frictionless crack front.
The above results show that the present ana- the initiation speed for the 5, 10 and 20 m/s impact
lyses predict intersonic crack propagation in the velocity remains below the shear wave speed and
unidirectional fiber composite when the impact the crack enters the intersonic regime when the
velocity exceeds a given threshold. At the lowest 5 impact velocity reaches 20 m/s. On the contrary,
m/s impact velocity, the crack accelerates after the the initiation speed at 30 and 40 m/s impact ve-
onset of extension and attains a velocity of about locities is above the Rayleigh wave speed (1811 m/
1465 m/s. This state is not a steady state and the s) and 94% and 110% of the shear wave speed of
crack speed first starts reducing. Then, it again the material, respectively. The maximum steady-
starts increasing gradually and attains the steady state crack speed is found to be 7600 m/s, for both
state at 1856 m/s, which is 95% of the shear wave impact velocities, which is 3.9 times the shear wave
speed (1949 m/s) of the material. At this impact speed and 83% of the longitudinal wave speed of
velocity, the crack tip deaccelerates before reach- the material.
ing the end of the allowable path. At 10 m/s impact The above results show that crack extension
velocity, the crack tip accelerates after initiation, speed and maximum crack speed, in sub-sonic re-
and when it attains a maximum speed. Basically, gime, monotonically increase with the increase in
the crack reaches a steady state with an average impact energy. The impact velocity of 10 m/s is
speed of 1879 m/s, which is 96% of the shear wave found to be the limit beyond which the dynamic
speed. Fig. 12 shows a different behavior when the crack propagation becomes intersonic. The crack
impact velocity is 20 m/s. After initiation, the attains a steady state in the intersonic regime and
crack speed remains almost steady at 766 m/s for the maximum speed is approximately the same for
10 ls and then accelerates to a speed of 2068 m/s, the three impact velocities of 20, 30 and 40 m/s as
thus crossing the shear wave speed threshold of the considered in the analyses. Thus, the possibility of
material. Then, it accelerates once more and at- intersonic crack propagation in unidirectional fi-
tains a near steady-state speed of 7512 m/s. Thus, ber composites is here established beyond an im-S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 497
pact velocity of 10 m/s for assumed steel impactor, ls. At this instant, a release wave as well as a
50 mm in diameter, and composite properties. The tensile transverse wave are generated. This release
crack speed attains a maximum of 83% of the wave unloads the material while propagating in
longitudinal wave speed, as the impact velocity or the þX-direction. It reaches the pre-notch tip at
imparted energy, exceeds a threshold. These fea- 30.1 ls and loads it further in shear due to the
tures are in close agreement with the analytical presence of the traction-free pre-notch surfaces.
predictions of Huang et al. (1999) in which a Crack extension depends on the history of the
critical crack velocity is identified, in the steady- energy delivered to the pre-notch tip. At the lowest
state regime, as the only velocity for which the impact velocity of 5 m/s, the energy imparted to
energy delivered to the crack tip is non-vanishing the tip by the incident wave is not sufficient until
and finite. The stress wave phenomena influencing the release tensile transverse wave reaches it and
crack extension and dynamic propagation is dis- starts loading the crack in mode I. At the highest
cussed next. impact velocities of 40 m/s, the shear energy im-
parted to the pre-notch tip is such that crack ex-
5.2.2. Stress wave history tension occurs earlier. The crack extends at 31.2 ls
The pre-notch tip stress and deformation fields, in predominantly mode II.
onset of crack extension and observed variation in
crack speed are here discussed in terms of stress
wave propagation within the plate. Upon asym- 5.2.3. Features of sub-sonic crack propagation
metric impact, compressive, as well as shear stress Fig. 13 shows crack tip state parameters, for an
waves, are generated in the lower half of the plate. impact velocity of 10 m/s, immediately after the
The waves travel in the X-direction towards the onset of crack extension and during crack propa-
pre-notch tip. The shape of the two stress waves gation at maximum speed. The plots are contours
are similar to the prescribed boundary velocity, of longitudinal stress in the fiber direction, S11 ,
i.e., the stress magnitude increases linearly from transverse (normal) stress S22 , shear stress SP12 ,
p p . Black ar-
zero to a maximum. The speed of the stress wave and effective plastic strain Eeff , i.e., E
normal to the fiber direction, i.e., in the Y-direc- rows show the position of the crack tip. The
tion, is 2966 m/s. This is approximately one third maximum crack speed in this case is 1879 m/s,
of the stress wave speed in the fiber direction. As a which is 96% of the shear wave speed of the
result, the compressive stress wave develops a composite. The figure shows that at the onset of
front of an expanding cylinder, instead of being crack extension, the normal traction is tensile and
spherical as it is the case in homogeneous materi- the tangential tractions across the pre-notch tip is
als. The longitudinal stress wave in the fiber di- oppositely directed making the shear stress con-
rection carries with it a compressive transverse tinuous across the crack front. Hence, the crack
stress and shear stress generated due to material initiates under mixed-mode loading. As shown
PoissonÕs effect and elastic anisotropy. These later, the normal and tangential tractions at the
stresses, specially the shear stress, appear to have a onset are of the same order of magnitude during
waveform traveling at the speed of the longitudinal mixed-mode loading. At steady state, the normal
wave speed in the fiber direction. Nevertheless, it traction above the crack front is compressive,
should not be confused with the shear wave, which which unloads the mode I loading, while the shear
travels at a much smaller speed of 1949 m/s. stress is continuous ahead of the crack tip. This
The compressive stress wave reaches the pre- suggests that the crack propagates at steady state
notch tip at 24.6 ls. The entire pre-notch tip is in predominantly mode II, which is confirmed in a
loaded in compression at this instant. Moreover, later discussion of interface traction history. The
the tip is also loaded in shear, i.e., mode II, due to figure also shows that the area of plastic flow in the
the material elastic anisotropy and asymmetric material is small and limited to the crack front.
impact loading. The compressive stress pulse The stress field does not present any discontinuity
reaches the traction-free left plate surface at 27.4 line behind the crack tip.498 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
p
Fig. 13. Contours of S11 , S22 , SP12 and Eeff for the case of 10 m/s impact velocity at onset of crack extension (top) and at steady state of
maximum crack speed (bottom).
5.2.4. Features of intersonic crack propagation steady-state intersonic crack propagation takes
The crack tip field changes in the intersonic place under predominantly mode II loading. The
regime. Figs. 14 and 15 show crack tip parameters plastic region remains small and limited to the
for impact velocities of 30 and 40 m/s at which the crack front and its wake. Most importantly, a
crack propagates at intersonic speed. In both shock wave resulting from the crack propagating
cases, the crack tip normal traction is compressive at a speed higher than shear and surface waves, are
at the onset of crack extension. The release wave seen emanating from the crack tip at 49 and 43 ls
generated from the left boundary has not reached for the 30 and 40 m/s impact velocity, respectively.
the crack tip, as can be inferred from the figures. The The shock waves are shown more clearly in Fig. 16
tangential traction is oppositely directed while the for the two impact velocities. In the figure, the
shear stress is continuous across the interface shock wave is marked by white dashed lines.
around the tip. Interface elements fail predomi- When the impact velocity is 30 m/s, the onset of
nantly under mode II loading. As the crack speed crack extension occurs at 32.4 ls predominantly in
increases and attains steady state, the release pulse shear. At this instant, the release wave, in the
catches up and the crack tip develops tensile nor- lower half plate, is reaching the crack front and the
mal traction, though small in magnitude. The upper half plate, near the crack tip, is under
phenomena increases the mixed-mode loading seen compression. The crack tip and its surrounding
by the reduction in the continuity of shear stress remain under compression. The crack accelerates
across the interface. The crack propagates pre- and attains maximum speed at steady state under
dominantly in shear, mode II. This shows that primarily mode II loading. As the release wave, in
though the crack tip loading is mixed-mode, the the lower half plate, catches up, the crack tipS.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509 499
p
Fig. 14. Contours of S11 , S22 , SP12 and Eeff for the case of 30 m/s impact velocity at onset of crack extension (top) and at steady state of
maximum crack speed (bottom).
normal loading changes to tensile. This change in formation of shock waves and their features are
mixed-mode loading coincides with oscillations in strongly dependent on the interface properties and
crack speed as can be seen in Fig. 12. However, plate dimensions. It should be noted that formation
these oscillations are small and the propagation of two shock waves were also experimentally ob-
speed can be assumed to be steady at its maximum served (Coker and Rosakis, 2001), as a function of
value. A single shock wave, manifested as a stress impact velocity.
discontinuity, develops behind the propagating A similar phenomenon occurs when the impact
crack tip in the upper half. This is evidently no- velocity is 40 m/s. The release wave in the upper
ticeable in the normal stress contours at 49 ls and half plate entraps the compressive stress field pre-
also noticeable in the tangential and shear stress sent at the crack front. The interaction of this re-
contours. Fig. 16 shows that the line of disconti- lease wave with the compressive field creates an
nuity emanates from the crack tip. The shock wave intense shear stress field ahead of the crack tip. A
trails the crack tip preserving its magnitude. The stress discontinuity develops, as seen in Figs. 15
inclination of the discontinuity with respect to a and 16, when the crack propagates intersonically.
vertical line remains constant during steady-state The inclination of the discontinuity line, from the
propagation as expected. The prediction of a single vertical line, is slightly larger than the inclination
shock wave is in contrast to our earlier analyses, for the 30 m/s impact velocity.
Dwivedi and Espinosa (2001), in which two dis-
continuity lines, one emanating from the crack tip 5.2.5. Nature of mode mixity with increased impact
and another behind it, were predicted for a 125 velocity
mm wide plate and an interface with a shear The normal and tangential traction histories at
strength of 300 MPa. This difference suggests that the center of the interface elements along the crack500 S.K. Dwivedi, H.D. Espinosa / Mechanics of Materials 35 (2003) 481–509
p
Fig. 15. Contours of S11 , S22 , SP12 and Eeff for the case of 40 m/s impact velocity at onset of crack extension (top) and at steady state of
maximum crack speed (bottom).
Fig. 16. Shock wave trailing behind the crack tip propagating at intersonic speed for the 30 m/s (left) and 40 m/s (right) impact ve-
locities.
path are shown in Fig. 17, for impact velocity of 5 tractions are plotted at an interval of 20 interface
m/s, and in Fig. 18, for impact velocities of 30 m/s elements, i.e., element numbers 1, 21, 41, etc. At 5
(left) and 40 m/s (right). Starting from the first m/s impact velocity, the crack extension occurs
element at the pre-notch tip (plotted in black), the under mixed-mode loading. This is seen by theYou can also read
