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On dynamic buckling phenomena in axially loaded elastic–plastic cylindrical shells
International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
On dynamic buckling phenomena in axially loaded elastic–plastic cylindrical shells D. Karagiozovaa , Norman Jonesb; ∗ a Institute
of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 4, So a 1113, Bulgaria Research Centre, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
b Impact
Abstract Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range in/uence the initial instability pattern and the 0nal buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t = 0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic elastic–plastic buckling; Cylindrical shells; Axial impact; Stress waves
1. Introduction The dynamic elastic–plastic buckling of structures is a complex phenomenon, due to various factors such as inertia e8ects, large deformations, material inelastic behaviour, etc. Due to the complexity of this phenomenon, it is not possible to obtain analytical solutions in the general case, so that simpli0ed models combining some preliminary analytical work together with further numerical study have been reported in the literature. These models clarify the e8ects of various parameters, which in/uence the buckling of axially loaded structural elements. Some studies have explored various aspects of the dynamic ∗ Corresponding author. Tel.: +44-151-497-4858; fax: +44-151-497-4848. E-mail address: [email protected] (N. Jones).
elastic response [1,2], while others have examined the static rigid plastic behaviour [3]. Further studies on dynamic elastic–plastic behaviour of simple models [4 –9] have shown that the elastic–plastic material properties introduce considerable complexity in the dynamic response of axially loaded structures. A discussion on some characteristic features of the dynamic elastic–plastic buckling phenomenon is the focus of this article. Of particular interest is the circular cylindrical shell, a simple structural element, whose response to a dynamic axial load exhibits many of the features, which are characteristic of the dynamic buckling phenomenon [10]. The experimental [11–15] and some recent theoretical studies [16 –19] show that the variety of the dynamic buckling responses of axially loaded shells are caused by the coupling of the inertia and inelastic material properties. In particular,
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 1 4 6 - 9
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Nomenclature c ce ; cp h vx ; vr u; w x; ; z D E Eh L P(t)
stress wave speed elastic and plastic stress wave speeds, respectively wall thickness of the shell axial and radial velocities, respectively axial and radial displacements, respectively axial, circumferential and radial directions, respectively diameter of a shell Young’s modulus strain hardening modulus in true stress–true strain space length of a shell axial force acting on a shell
the three types of instability namely dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements), dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) and global bending (when the shell buckles in an Euler mode) develop according to the geometry of the shell, loading conditions and the material properties [11–15]. Thus, it is important to identify the factors causing the transition between the types of buckling and to determine the load causing a structural collapse. The principal dynamic e8ects on the elastic–plastic buckling behaviour of axially loaded cylindrical shells and many other thin walled structural elements are associated with the lateral inertia. The in/uence of axial inertia is neglected usually in theoretical studies. On other hand, the in/uence of the material properties on dynamic buckling is analysed mainly from the viewpoint of strain rate e8ects, which strengthen a material, but less attention is paid to the in/uence of the actual stress–strain relationship in the plastic range and to the approximation of this curve on the dynamic buckling predictions. A linear approximation of this relationship is assumed in some theoretical
Pmax R T0 Tc V0 x ;
xp ; p ; ep 0 x ; ; e
maximum load at t = 0 applied to the proximal end mean radius of a shell initial impact energy energy absorbed by a shell during the initial stable response initial velocity axial and circumferential total strains, respectively axial, circumferential and equivalent plastic strains, respectively material density Poisson’s ratio yield stress axial, circumferential and equivalent stresses, respectively
studies [4 –9,11,12,16,17], while non-linear approximations are considered usually in numerical analyses. In the particular case of an axially loaded cylindrical shell, sequential phases of compression and bending govern the deformation process, which form a complex loading=unloading path. Therefore, the phenomenon of dynamic buckling is very sensitive to the loading path in the plastic range, as observed in Ref. [4] for an idealised elastic–plastic model. From this viewpoint, it is important to analyse the in/uence of the material approximations on the buckling response. It appears, that the character of the initial stable phase of a shell deformation, when axial compression dominates the shell motion, in/uences the instability pattern, which develops later. Factors, which cause a transition from dynamic progressive to dynamic plastic buckling, have been identi0ed in [17–19] and they are closely associated with the material characteristics and the inertia properties of a shell. The purpose of this study is to explore the in/uence of approximations in the inelastic properties of a material on the predicted pattern of axisymmetric buckling and the in/uence of the axial inertia on the initiation and development of buckling of circular cylindrical shells subjected to a dynamic axial load.
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An axial impact on a stationary cylindrical shell is modelled as shown in Fig. 1 when varying the initial velocity of the striking mass between 10 and 75 m=s. All shells have a diameter D = 35 mm and a thickness h = 1:5 mm, and three lengths L = 3D; 4D or 5D. It is assumed that the shells are made of an aluminum alloy 6063-T5 [20] with elastic modulus E = 69 GPa, density = 2685 kg=m3 , yield stress 0 = 172 MPa and the actual static stress–strain relation in Fig. 2. The von Mises yield criterion is assumed with isotropic strain hardening. Material strain–rate e8ects are taken into account in some material models. Several multi-linear approximations of the actual stress–strain curve are considered in order to analyse their in/uence on the dynamic buckling phenomenon and these approximations are presented in the true stress–true strain space in Fig. 2. Two values for the hardening modulus, Eh , are used for the approximations 1 and 2 giving a strain energy equivalent to the
true stress, MPa
2. Shell geometry, material approximation and loading conditions 300
150
....... Approximation A1 - - - - Approximation A2 ____ Approximation A3
0 0.0
0.2
0.4
0.6
true strain Fig. 2. Material approximations. • • • Actual stress–strain relationship.
strain energy based on the actual stress–strain curve at 10% and 20% true strain, respectively. A hardening modulus of 880 MPa is assumed for the equivalent strain, ep , between 0 and 0.1, but Eh = 200 MPa for ep ∈ (0:1; 0:5) for approximation A1. A hardening modulus of 605 MPa is assumed for the equivalent strain, ep , between 0 and 0.2, but Eh = 156 MPa for ep ∈ (0:2; 0:5) for approximation A2. A piecewise linear presentation is used in approximation A3 when hardening moduli of 1072, 752, 448 and 250 MPa are associated with the true strain ranges of (0, 0.025), (0.025, 0.05), (0.05, 0.1) and (0.1, 0.5), respectively. Approximations 4 and 5 take into account material strain rate e8ects according to the Cowper–Symonds equation q e ep = D 0 − 1 (1) with D = 1; 288; 000 s−1 ; q = 4 [10] when using the corresponding static curves from approximations A2 and A3. In Eq. (1), 0 (ep ) is the static /ow stress, which depends on the plastic strain via isotropic hardening. 3. Modelling of the shell impact
Fig. 1. Shell geometry and the co-ordinate system.
A numerical simulation of the impact event was carried out using the FE code ABAQUS=Standard, version 5.8. It is assumed that an axisymmetric buckling mode develops in suNciently thick shells [12,13],
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so that 2D axisymmetric 8 nodes biquadratic solid elements are used in the simulation. The load is applied as a point mass attached to the nodes of a rigid body, which has an initial velocity V0 . All shells are modelled with three elements across the shell thickness while 60, 80 and 100 elements are used along the shells having L = 3D; 4D or 5D, respectively. The contact between the shell and the striker is de0ned using the ‘surface interaction’ concept together with a friction coeNcient 0.25 at the proximal end. The self-contact of the inner and the outer surfaces during the deformation of the shell are assumed frictionless. The axial and the radial degrees of freedom, u; w, of the nodes at the distal end are 0xed in order to model a clamped end, but no constraints are assumed for the degrees of freedom associated with the nodes at the proximal end when modelling a clamped-free shell. The modelled shells have no initial imperfections.
Fig. 3. Patterns of shell instability for a shell having L = 5D; V0 = 75 m=s; T0 = 2 kJ. (a) Dynamic progressive buckling—t = 0:356 ms, material approximation A3. (b) Dynamic plastic buckling—t = 0:324 ms, material approximation A2.
4. Patterns of buckling Several experimental studies on the axial impact of elastic–plastic cylindrical shells show that the response of a cylindrical shell, which buckles inelastically develops in two stages: initial instability pattern dominated by axial compression and a structural collapse when large deformations occur [20]. Recent studies reported in the literature [17–19] reveal that the initial instability pattern for elastic–plastic cylindrical shells subjected to an axial impact loading depends not only on the inertia characteristics of the shell, but is strongly in/uenced by the material hardening properties. Dynamic progressive buckling (Fig. 3(a)) dominates the shell response for a wide range of loading and material parameters [13–15], while dynamic plastic buckling (Fig. 3(b)) can develop for high velocity impacts and particular material properties [12,17–19]. It was observed in [17–19] for a particular cylindrical shell with a bilinear stress–strain approximation that the ratio between the material strain hardening and elastic moduli governs the initial pattern of instability. Dynamic plastic buckling is characterised by an initial wrinkling within a sustained plastic /ow along the shell length during the 0rst phase of deformation [12,17–19], while the strain localises in the case of progressive buckling [18,19].
A comparison between the initial shell instability and the 0nal buckling shapes of a shell having a length L = 4D and subjected to a 2 kJ impact applied with V0 =75 m=s is presented in Fig. 4 for di8erent material approximations. It is evident that the material approximations in/uence signi0cantly the buckling shapes. Dynamic plastic buckling characterises the initial instability pattern for a shell with the material properties approximation A2, while an initial instability in the dynamic progressive pattern is associated with the approximations A3 and A4. On other hand, the material approximations are more signi0cant when predicting the response to a high velocity impact, as shown in Fig. 5. It is evident that the buckling shapes di8er considerably for impact velocities higher than 30 m=s when using material approximations A1 and A3. The buckling patterns presented in Figs. 4 and 5 show that the approximations in the material properties together with the inertia e8ects in/uence the pattern of shell instability when assuming axisymmetric buckling. 5. Stress wave propagation The axial inertia of a shell plays an important role for higher impact velocity and has a signi0cant
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Fig. 4. Initial shell instability and the 0nal buckling shapes of shells having L = 4D; V0 = 75 m=s; T0 = 2 kJ. (a) Initial, (b) approximation A2, t = 0:235 ms, (c) approximation A2, t = 2:83 ms, (d) approximation A3, t = 0:235 ms, (e) approximation A3, t = 2:997 ms, (f) approximation A4, t = 0:235 ms, (g) approximation A4, t = 2:540 ms.
e8ect on the buckling initiation patterns. A high velocity impact causes an instantaneously applied force on the shell and initiates elastic and plastic stress waves, which propagate along the shell. It was shown in [17,18] that the assumption of a biaxial stress state, (x = 0; = 0), can be used with suNcient accuracy to analyse the transient deformation process in circular cylindrical shells. A plastic wave can propagate at di8erent speeds in shells depending on the assumed material model. The governing equations of motion of an elastic–plastic medium in a biaxial stress state (x = 0; = 0) are [18]: At wt + Ax wx + b = 0; where
0 At = 0 0
0 0
0 0 0 ; R 0 0
0 0 b= ; − −w=R ˙
(2)
0 −1 Ax = 0 0
−1 0 0 0
0 0 0 0
˙ ] T ˙ x ; w; w = [ u;
0 0 ; 0 0
and were used to obtain the speeds of the elastic and plastic waves, which can propagate in axially loaded cylindrical shells. The plastic stress waves in shells made of a strain rate sensitive material have a constant speed c = ±(E=(1 − 2 ))1=2 = cp ;
(3)
when the strain–rate e8ects are taken into account according to the Cowper–Symonds equation [19]. The stress wave speeds for strain rate insensitive materials depend on the stress state in the plastic range and e.g. for a von Mises material the wave speeds can be expressed as [18,19]
1=2 c=± ; (4) ( − 2 ) where (2x − )2 1 ; + g (e ) = E (2e )2 (2x − )(2 − x ) = − + g (e ) ; E (2e )2 1 (2 − x )2 = + g (e ) E (2e )2
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D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238 σθ/σ0 p
2
e
c /c
1
Yield locus
0 0.1 -1
-λ = 0.01
−λ = 1 (elastic)
-2
-2
-1
0
1
2 σx/σ0
Fig. 6. Stress wave speeds in isotropic elastic–plastic materials with a linear strain hardening and the von Mises yield criterion ( = 1; 0:1; 0:01).
Fig. 5. Final buckling shapes of shells having L = 3D subjected an impact with T0 = 1:5 kJ for material approximation A1 (b–e) and material approximation A3 (g–j); (b,g) V0 = 10 m=s, (c,h) V0 = 30 m=s, (d,i) V0 = 60 m=s, (e,j) V0 = 75 m=s.
ep
and where = g(e ) is the stress–strain relationship in the true stress–true strain space with g (e ) = (1 − )=E and = Eh =E for a material with linear strain hardening, Eh . An illustration of the stress wave speed dependence on the stress state (x ; ) and for di8erent hardening ratios , are presented in Fig. 6 as a unit vector (cp =ce ) perpendicular to the initial yield locus for the von Mises yield condition ( varies between 0.0036 and 0.0115 for the approximations of Al 6063-T5 in Section 2). It is evident that the elastic wave speed is an upper bound for plastic wave speeds in a strain rate insensitive material. The minimum values for the plastic wave speeds show a strong dependence on the hardening modulus, which can be compared to the corresponding proportionality to the hardening
modulus in the uniaxial case. It is shown in [18], however, that the minimum values for the plastic wave speed in strain rate insensitive materials are always larger than the plastic wave speed in a uniaxial medium as p p p cmin; Mises ¿ cmin; Tresca ¿ cuniaxial ;
(5)
p p where cmin; Mises and cmin; Tresca are the minimum values for the plastic wave speeds, which propagate in shells made of materials obeying the von Mises or Tresca yield criteria. For a bilinear elastic–plastic material the corresponding expressions for the plastic wave speeds in circular shells are obtained as [17–19] p 1=2 2 cmin; Mises = ±(E=) {4 [4 (1 − )
+3(1 − )]−1 }1=2
(6)
for ( =x =2; x ¡ 0; ¡ 0) and (( =x =2; x ¿ 0; ¿ 0), p cmin;
Tresca
= ±(E=)1=2 {2 [2 (1 − 2 ) √ +(1 − ) 3]−1 }1=2
(7)
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Fig. 7. Development of the buckling shape for a shell with L = 4D; material approximation A2, V0 = 75 m=s, T0 = 2 kJ; (a) initial, (b) t = 0:217 ms, (c) t = 0:557 ms, (d) t = 0:735 ms, (e) t = 0:951 ms, (f) t = 1:176 ms, (g) t = 1:83 ms, (h) t = 2:83 ms.
Fig. 8. Development of the buckling shape for a shell with L = 4D; material approximation A3, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:233 ms, (c) t = 0:638 ms, (d) t = 0:963 ms, (e) t = 1:2 ms, (f) t = 2:54 ms, (g) t = 2:977 ms.
for ( = x ; x ¡ 0; ¡ 0) and ( = x ; x ¿ 0, ¿ 0), while the uniaxial plastic wave speed is p cuniaxial = ±( E=)1=2 :
(8)
The development of the buckling shape with time of a shell having a length L = 4D subjected to a
75 m=s axial impact is presented in Figs. 7–10 for the material approximations A2, A3, A4 and A5, respectively. It is evident that di8erent buckling patterns are associated with the various approximations. The axial inertia of the shell plays a signi0cant role for such a high velocity impact so that the transient deformation process becomes an
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Fig. 9. Development of the buckling shape for a shell with L = 4D; material approximation A4, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:235 ms, (c) t = 0:480 ms, (d) t = 0:732 ms, (e) t = 1:018 ms, (f) t = 2:54 ms.
Fig. 10. Development of the buckling shape for a shell with L = 4D; material approximation A5, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:235 ms, (c) t = 0:340 ms, (d) t = 0:792 ms, (e) t = 1:66 ms, (f) t = 2:243 ms.
important part of the response. Figs. 11(a) – (c) present the axial and the circumferential strains along the shell mid-surface at several times for three of the material models. The axial strain distribution at t = 0:217 ms in Fig. 11(a) (approximation A2) shows that the shell has buckled with wrinkles along the
entire length (dynamic plastic buckling, Fig. 7(b)). The particular impact velocity of 75 m=s causes instantaneous plastic deformations at the proximal end of the shell and a plastic wave having a speed of approximately 550 m=s. The primary plastic wave and the secondary one resulting from the
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strains 0.1
εθ
0.235 0.217
0.0
0.144 0.055 msec 0.144
0.217
-0.1
0.100
0.235
-0.2
(a)
εx
0.000
0.035
0.070
0.105
0.140
x, m
strains εθ
0.1 0.0
0.074 msec
/ 0.181
0.215
-0.1
/ 0.154
/ 0.107
-0.2
(b)
0.000
εx
0.035
0.070
0.105
0.140
x, m strains 0.1
εθ
0.235
0.0
0.050 msec 0.190
0.100
0.07
-0.1 0.235
-0.2
0.000 (c)
εx
0.035
0.070
0.105
0.140
x, m
Fig. 11. Strain distribution until the initiation of buckling in a shell having L = 4D and di8erent material approximations; V0 = 75 m=s; T0 = 2 kJ. (a) material approximation A2, (b) material approximation A3, (c) material approximation A4.
re/ection of the elastic wave from the distal end meet at t ≈ 0:140 ms when the entire length of the shell is under large axial compression and until
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t ≈ 0:190 ms buckling develops within a sustained plastic /ow. High velocity impacts on a shell made of an elastic– plastic material with approximation A3 cause plastic waves to propagate at di8erent speeds with the largest strains travelling slowest [21]. The strain patterns in Fig. 11(b) (approximation A3) show that a 75 m=s velocity impact causes instantaneous axial compressive strains of approximately 0.12, which is associated with Eh = 250 MPa. The plastic wave emanating from the proximal end and having a lower speed (approximately 350 m=s) can propagate only a short distance (≈ 24 mm) before the radial displacements near this particular end start to grow rapidly (Fig. 8(b)) for t ¿ 0:07 ms since the radial inertia cannot any longer support the unbuckled shape. The elastic unloading across the shell thickness interrupts the further propagation of the plastic stress wave so that a localisation of strains occur near the proximal end of the shell causing a further progressive folding, although small strains at higher speeds continue to propagate. The axial strains along the shell with material approximation A4, which includes material strain rate e8ects, are presented in Fig. 11(c). In this case, the plastic waves propagate at the elastic wave speed and the axial strains, although they vary in magnitude along the shell, increase with time along the entire length. Similarly to approximation A2 in Fig. 11(a), buckling occurs within a sustained plastic /ow, however even when the stresses start to decrease for the visco-plastic material, the strains can continue to increase [19]. This material behaviour causes a strain localisation near to the shell ends (Fig. 11(b)) and further progressive buckling. It is evident for material approximation A5 in Fig. 10(b) that the strain localisation at both ends of a shell is even more signi0cant than for material approximation A4. This phenomenon occurs because the static stress–strain curve for approximation A5 has lower strain hardening properties than material approximation A4. The development of the buckling shapes presented in Figs. 7–10 reveals that regardless of the initial pattern of the shell instability, dynamic buckling always continues progressively provided suNcient external energy is available. The initiation of buckling, however, a8ects the 0nal buckling shapes as shown in Fig. 4.
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6. Axial inertia e&ects The responses of shells having di8erent lengths are examined in order to explore further the in/uence of the axial inertia on the initial instability pattern and the 0nal buckling shape. The 0nal buckling shapes of shells having L = 5D; 4D and 3D and subjected to 2 kJ impact with an initial velocity 75 m=s are presented in Fig. 12 for material approximations A3 (Fig. 12(b),(f),(j)), A2 (Fig. 12(c),(g),(k)) and A1 (Fig. 12(d), (h),(l)). It is evident that the number of the developed folds depends on the material approximation but not on the shell length. However, the location of the folds depends on the shell length for shells made of material A3. An explanation for this phenomenon is related to the axial inertia e8ects on the buckling mechanism. Material approximations A1 and A2 allow larger strains to propagate at higher speeds along the shell due to the larger hardening moduli (see Fig. 11(a)). For these material approximations, the plastic waves can propagate along the shell lengths (L=5D; 4D and 3D) causing dynamic plastic buckling without initial strain localisation near to the proximal end. A localisation of strains develops near to the distal end due to the re/ected stress wave causing further progressive folding near this particular end (see Fig. 7). Plastic stress waves for material approximation A3 propagate at di8erent speeds from the proximal end. The slower stress waves carry larger stresses and strains thus causing strains with di8erent magnitudes to propagate along the shell. In this case, the strain localisation near to the proximal end causes progressive folding to develop initially near this particular end as shown in Figs. 8 (b),(c) (L = 4D) and 13 (b),(c) (L = 3D). However, the re/ected stress wave from the distal shell end of the shell having L = 3D initiates the strain localisation and a rapid growth of the radial displacements more quickly due to the shorter length as evident when comparing Figs. 8 and 13. Two complete wrinkles near to the distal end characterise the 0nal buckling shape of the shell with L = 3D, while only one complete wrinkle develops in the shell having L = 4D and no complete wrinkle is observed for the shell of L=5D in Fig. 12(b). The piecewise linear material approximation can have an additional e8ect in connection with the rate
of loading. The development of the buckling shapes of a shell with L = 4D and a material approximation A3, which is subjected to a 1:5 kJ impact is presented in Figs. 14 and 15 for V0 = 60 and 30 m=s, respectively. The higher velocity impact causes larger plastic strains to develop near to the proximal end at the early stage of the shell response and these strains propagate relatively slowly. The radial inertia cannot support such high stresses and the shell starts to buckle near to the proximal end as evident in Fig. 14 (b),(c). Smaller plastic strains, which propagate at a higher speed, emanate from the proximal end as a result of a 30 m=s impact. These strains are too small to cause any initial buckling near to the proximal end, but the re/ected stress wave produces buckling near to the distal end, as shown in Fig. 15(b). One wrinkle develops near the proximal end at a much later time (Fig. 15(d)) and further progressive buckling forms the 0nal buckling shape. A 60 m=s impact also causes progressive buckling, however, the sequence of the wrinkle development is di8erent, as evident in Fig. 14. 7. Buckling load for high velocity impacts Unlike the elastic and elastic–plastic response to a low velocity impact, when the load increases from zero and buckling occurs upon reaching a critical value, the maximum load on a shell resulting from a high velocity axial impact occurs instantaneously and the associated strains exceed the elastic limit of the material. The peak load at t = 0 results from the axial stress discontinuity produced at t = 0 and is independent of the striking mass [19]. As a high velocity impact is characterised by a signi0cant radial inertia of the shell, the peak load may or may not cause buckling depending on the initial kinetic energy of the striker [19]. A small initial kinetic energy applied with a high impact velocity can be absorbed only by an axial compression and a uniform expansion of the tube, but larger energies can cause initial instability and a further collapse [18]. A relationship between the impact velocity and the maximum value of the applied load, which occurs at t = 0, can be obtained when analysing the stress wave propagation in an elastic–plastic shell [18,19]. The initial response of an elastic–plastic cylindrical shell to
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Fig. 12. Final buckling shapes of shells subjected to an impact with T0 = 2 kJ; V0 = 75 m=s (a–d) L = 5D, (e–h) L = 4D, (i–l) L = 3D, (b,f,j) approximation A3, (c,g,k) approximation A2, (d,h,l) approximation A1.
an axial impact is characterised by large axial compressive strains and negligible circumferential strains [18], which leads to ≈ x =2; x ¡ 0; ¡ 0 when using the von Mises yield criterion and an initiation p of plastic waves having speeds cp ≈ cmin . In the case of a piecewise linear stress–strain relationship (approximation A3), the instantaneous plastic strains depend on the impact velocity and they are
associated with the corresponding hardening modulus for di8erent parts of the stress–strain curve. Similarly to the case of a strain rate insensitive isotropic material with a linear hardening [18,19], the peak load, which acts on a shell made of an elastic–plastic material with a piecewise linear material approximation, is a function of the material properties and the impact velocity.
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Fig. 13. Development of the buckling shape for a shell with L = 3D; material approximation A3; T0 = 2 kJ; V0 = 75 m=s; (a) initial, (b) t = 0:302 ms, (c) t = 0:546 ms, (d) t = 0:751 ms, (e) t = 1:328 ms, (f) t = 2:323 ms, (g) t = 3:55 ms.
Fig. 14. Development of the buckling shape for a shell with L = 4D; material approximation A3; T0 = 1:5 kJ; V0 = 60 m=s; (a) initial, (b) t = 0:322 ms, (c) t = 0:504 ms, (d) t = 0:918 ms, (e) t = 1:598 ms, (f) t = 2:612 ms.
The di8erential relations along the characteristics of the problem determined by Eq. (2) lead to [18] dvx vr dx dx ∓ cp = − (cp )2 along = ±cp dt dt R dt (9) and vr d
dx = ; + dt dt R
(10a)
dx dvr = along = 0; (10b) dt dt which results in equation
2 (cp )2 d x + (cp )2 d = ±cp dvx : 1+ (11)
R
Using the initial condition vr = 0 at x = L;
(12)
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the stress discontinuity occurs in the nth linear portion of the stress–strain diagram having a hardening modulus (Eh )n , an expression for the axial stress, x|x=L , at t = 0 can be obtained as x|x=L n−1 p = %(x )i + (cmin )n
i=1
n−1 %(x )i V0 − p (cmin )i i=1
: (17)
p The terms %(x )i =(cmin )i in Eq. (17) can be denoted formally as velocity increments, (Tvx)i , so that the peak load, which can result from an axial impact on a shell made of a piecewise linear material (e.g. approximation A3), can be expressed as √ p Pmax; Mises =2&Rh = 20 = 3 + (cmin )n
Fig. 15. Development of the buckling shape for a shell with L=4D; material approximation A3; T0 = 1:5 kJ; V0 = 30 m=s; (a) initial, (b) t=0:765 ms, (c) t=2:506 ms, (d) t=3:762 ms, (e) t=5:672 ms.
the dynamic discontinuity condition on the wave front is obtained as p (x ; )vx = 0; dx − dcmin
i.e. the velocity on the wave front is
1 x d Sx vx = p : 0 cmin The above relationship can be presented as n n 1 (x )i+1 d Sx 1 %(x )i V0 = = p p (cmin )i (cmin )i (x )i i=1 i=1
(13)
(14)
(15)
for a piecewise linear material and initial condition p vx|x=L = V0 and where (cmin )i is the minimum plastic wave speed associated with the linear strain hardening modulus (Eh ). The piecewise linear axial stress, (x )i , can be expressed in the form √ (16) (x )i = 2(e )i = 3; where (e )i is the equivalent stress associated with the particular linear region of the stress–strain relation and when assuming ( )i = (x )i =2. If the impact velocity is high enough to cause instantaneous plastic deformations at the proximal end of the shell, then the peak load, Pmax; Mises , can be related to the stress discontinuity at t = 0. Assuming that the magnitude of
× V0 −
n−1
(Tvx )i
+
i=1
which reduces to [18] √ p x|x=L = 20 = 3 + cmin V0
n−1 i=1
p (cmin )i (Tvx )i
; (18a) (18b)
for a shell made of an elastic–plastic strain rate insensitive material with a single hardening modulus. The predictions for peak loads of 50.9 and 49:7 kN resulting from a 75 m=s impact on a shell with L = 4D when assuming the material approximations A2 and A3 (Eq. (17)) compare well with the corresponding numerical values of 50.8 and 49:2 kN in Figs. 16(a) and (b). Because of the similar magnitudes of the stress discontinuity at t = 0, the initial deformation phases for both material approximations are identical. Both force–time and force–displacement histories in Fig. 16 are characterised by a rapid decrease of the suddenly applied force. The associated axial compression of the shell is about 3 mm without any bending deformations. This phase lasts for about 0:035 ms and during this time the plastic waves travel at ∼550 m=s and propagate a distance of 20 mm, which is comparable to the critical wavelength of the shell (12:75 mm [22]). Thus, buckling develops within a sustained plastic /ow near to the proximal end of the shell. After this time the shell responses develop di8erently due to the di8erent hardening
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P, N 40000
20000
0 0.000
0.002
0.001
(a)
0.003
t, sec
P, N 40000
20000
0 0.000
(b)
0.002
0.001
0.003
t, sec P, N 40000
20000
0
0 0.00 (c)
length of the shell with the material approximation A2 due to the higher speed of the plastic wave travelling along the shell (Fig. 7(b)). In contrast, the lower plastic wave speeds associated with the larger strains for the material approximation A3 cause a further localisation of the strains near to the proximal end of the shell and local bending occurs (Fig. 8(b)). The force–time histories in Figs. 16(a) and (b) indicate that a shell made of a material approximated as A2 can support a shape with only small radial displacements up to t ≈ 0:24 ms and a rapid buckling occurs thereafter (Fig. 7). For t ¡ 0:24 ms, an almost constant force acts on the proximal end of the shell. By way of contrast, a shell with material approximation A3 responds by a sudden buckling (Fig. 8), which is manifested as a continuous decrease of the force acting on the impacted end. It is evident, that the initial stable response of a shell, which is characterised by an axial compression, lasts longer for shells, which are made of inelastic materials with high strain hardening properties. This behaviour is due to the plastic waves, which propagate at higher speeds in these materials and cause large axial compressive strains to develop along the entire length of a shell before any elastic unloading occurs across the shell thickness. Fig. 16(c) shows that a shell made from material A2 (solid curve) can compress signi0cantly (u|x=L ≈ 0:017 m) before the development of large radial displacements. In this case, the stress discontinuity at t = 0 can be used to estimate the upper value for the impact energy, Tc , which the shell can absorb during the initial stable compression phase as [10]
e0 e de ≈ 2&RhLe0 (0 + e0 )=2; (19) Tc = 2&RhL
0.02
0.04
0.06
0.08
crushing distance, m
Fig. 16. (a) Force–time history at proximal end for initial velocities 75 m=s; L=4D; material approximation A2. (b) Force–time history at proximal end for initial velocities 75 m=s; L = 4D; material approximation A3. (c) Force–displacement histories at proximal end for initial velocity 75 m=s (T0 = 2 kJ); L = 4D; —— material approximation A2, - - - - - - material approximation A3.
characteristics associated with the material approximations A2 and A3. Small wrinkles within a sustained plastic /ow continue to develop along the
when neglecting elastic e8ects and where e0 and e0 are the equivalent stress and the corresponding equivalent plastic strain at t = 0. The absorbed energy estimated by Eq. (19) is an upper bound for the energy absorbed during the initial compression phase, since the stress along the tube is smaller than the stress discontinuity at t = 0 at the proximal shell end. However, if it is assumed that the axial stress along the tube is equal to the stress discontinuity determined by, e.g. Eq. (18), then the equivalent plastic strain can be obtained as e0 = ep = (e0 − 0 )=Eh :
(20)
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
For example, a shell having a length L = 4D and made of a material approximated as A2 can absorb up to 0:8 kJ in axial compression when impacted at 75 m=s according to Eqs. (18) – (20) and (6). It can be expected that energies ¡ 0:8 kJ (corresponding to an impact mass of 0:284 kg for the initial velocity of 75 m=s) cannot cause an inelastic collapse of the shell, but only an axial compression with small wrinkles along the shell developed within a sustained plastic /ow, which do not cause a local shell instability. For example, the deformed shape of the shell, presented in Fig. 4(b) at t = 0:235 ms is associated with an absorbed energy of about 0:6 kJ at this particular time. Consequently, the maximum value of the dynamically applied force to a shell, which buckles inelastically, does not play the same role as the critical dynamic load for elastic shells because of the axial inertia e8ects and the energy dissipated by the inelastic material behaviour during the initial stable response of the shell. While the maximum dynamic load is critical for the elastic shells and depends only on the radial inertia of the shell, the maximum dynamic load applied to an inelastic shell can be considered as a critical buckling parameter only in relation to the applied impact energy and the proportion of this energy, which can be absorbed during the initial axial compression phase. 8. Conclusions The in/uence of dynamic e8ects in the elastic– plastic buckling phenomenon is considered from the viewpoint of a transient process in cylindrical shells subjected to an axial impact. It is shown that the initiation of buckling for a high velocity impact is sensitive to the material properties, particularly to the hardening modulus, and the plastic wave speed determined by this modulus can lead either to dynamic plastic buckling or to dynamic progressive buckling. The material approximation can change the predicted pattern for the initial instability and the 0nal buckling shape of a shell. A regular buckling shape can occur for strain rate insensitive materials when buckling develops within a sustained axial plastic /ow, while the localisation of strains leads to progressive buckling in shells made of materials having a low strain hardening behaviour or strain rate sensitive properties. The
1237
rate of loading causes an additional e8ect on the buckling shapes of the shells made from an elastic–plastic material with piecewise linear strain-hardening characteristics because the stresses then propagate at different wave speeds along a shell. It is shown that the local inelastic instability of a shell (a structural collapse) subjected to a high velocity impact depends not only on the magnitude of the applied load, but also on the proportion of the initial kinetic energy, which can be absorbed by the axial compression of a shell during the initial stable response.
Acknowledgements The work outlined in this paper was supported by EPSRC Grant GR=M=750044.
References [1] D.A. Danielson, Dynamic buckling loads of imperfection– sensitive structures from perturbation procedure, AIAA J. 7 (1969) 1506–1510. [2] A.N. Kounadis, Nonlinear dynamic buckling of discrete structural systems under impact loading, Int. J. Solids Struct. 30 (1993) 2895–2909. [3] J.M. Hutchinson, On the post-buckling behaviour of imperfection–sensitive structures in the plastic range, J. Appl. Mech. 39 (1972) 155–162. [4] N. Jones, H.L.M. dos Reis, On the dynamic buckling of a simple elastic–plastic model, Int. J. Solids Struct. 16 (1980) 969–989. [5] D. Karagiozova, N. Jones, Dynamic buckling of a simple elastic–plastic model under pulse loading, Int. J. Non-linear Mech. 27 (1992) 981–1005. [6] D. Karagiozova, N. Jones, Some observations on the dynamic elastic–plastic buckling of a structural model, Int. J. Impact Eng. 16 (1995) 621–635. [7] X.Y. Su, T.X. Yu, S.R. Reid, Inertia-sensitive impact energy absorbing systems. Part 1. E8ects of inertia and elasticity, Int. J. Impact Eng. 16 (1995) 651–672. [8] X.Y. Su, T.X. Yu, S.R. Reid, Inertia-sensitive impact energy absorbing systems. Part 2. E8ects of strain rate, Int. J. Impact Eng. 16 (1995) 673–689. [9] D. Karagiozova, N. Jones, Multi-degrees of freedom model for dynamic buckling of an elastic–plastic structure, Int. J. Solids Struct. 33 (1996) 3377–3398. [10] N. Jones, Structural Impact, Paperback Edition, 1997, Cambridge University Press, Cambridge, 1989.
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[11] W. Abramowicz, N. Jones, Transition from initial global bending to progressive buckling of tubes loaded statically and dynamically, Int. J. Impact Eng. 19 (1997) 415–437. [12] A.L. Florence, J.N. Goodier, Dynamic plastic buckling of cylindrical shells in sustained plastic /ow, J. Appl. Mech. 35 (1968) 80–86. [13] K. Murase, N. Jones, The variation of modes in the dynamic axial plastic buckling of circular tubes, in: N.K. Gupta (Ed.), Plasticity and Impact Mechanics, Wiley Eastern Limited, New Delhi, 1993, pp. 222–237. [14] Li Ming, Wang Ren, Han Mingbao, An experimental investigation of the dynamic axial buckling of cylindrical shells using a Kolsky bar, Acta Mech. Sinica 10 (1994) 260–266. [15] T. Kurokawa, T. Sasaki, H. Suzuki, Y. Iamada, Axisymmetric collapse of circular tubes due to longitudinal impact, Proceedings of the Sino-Japanese Symposium on Deformation=Fracture of Solids, Huangshan= China, 1997, pp. 9 –16.
[16] D. Karagiozova, N. Jones, Dynamic elastic–plastic buckling phenomena in a rod due to axial impact, Int. J. Impact Eng. 18 (1996) 919–947. [17] D. Karagiozova, N. Jones, Dynamic elastic–plastic buckling of circular cylindrical shells under axial impact, Int. J. Solids and Struct. 37 (2000) 2005–2034. [18] D. Karagiozova, M. Alves, N. Jones, Inertia e8ects in axisymmetrically deformed cylindrical shells under axial impact, Int. J. Impact Eng. 24 (2000) 1083–1115. [19] D. Karagiozova, N. Jones, In/uence of stress waves on the dynamic plastic and dynamic progressive buckling of cylindrical shells under axial impact, Int. J. Solids Struct. 38 (2001) 6723–6749. [20] H.E. Lindberg, A.L. Florence, Dynamic Pulse Buckling, Martinus Nijho8, Netherlands, 1987. [21] W.K. Nowacki, Stress Waves in Non-Elastic Solids, Pergamon Press, UK, 1978. [22] W. Abramowicz, N. Jones, Dynamic axial crushing of circular tubes, Int. J. Impact Eng. 2 (1984) 263–281.
On dynamic buckling phenomena in axially loaded elastic–plastic cylindrical shells D. Karagiozovaa , Norman Jonesb; ∗ a Institute
of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 4, So a 1113, Bulgaria Research Centre, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
b Impact
Abstract Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range in/uence the initial instability pattern and the 0nal buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t = 0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic elastic–plastic buckling; Cylindrical shells; Axial impact; Stress waves
1. Introduction The dynamic elastic–plastic buckling of structures is a complex phenomenon, due to various factors such as inertia e8ects, large deformations, material inelastic behaviour, etc. Due to the complexity of this phenomenon, it is not possible to obtain analytical solutions in the general case, so that simpli0ed models combining some preliminary analytical work together with further numerical study have been reported in the literature. These models clarify the e8ects of various parameters, which in/uence the buckling of axially loaded structural elements. Some studies have explored various aspects of the dynamic ∗ Corresponding author. Tel.: +44-151-497-4858; fax: +44-151-497-4848. E-mail address: [email protected] (N. Jones).
elastic response [1,2], while others have examined the static rigid plastic behaviour [3]. Further studies on dynamic elastic–plastic behaviour of simple models [4 –9] have shown that the elastic–plastic material properties introduce considerable complexity in the dynamic response of axially loaded structures. A discussion on some characteristic features of the dynamic elastic–plastic buckling phenomenon is the focus of this article. Of particular interest is the circular cylindrical shell, a simple structural element, whose response to a dynamic axial load exhibits many of the features, which are characteristic of the dynamic buckling phenomenon [10]. The experimental [11–15] and some recent theoretical studies [16 –19] show that the variety of the dynamic buckling responses of axially loaded shells are caused by the coupling of the inertia and inelastic material properties. In particular,
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 1 4 6 - 9
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Nomenclature c ce ; cp h vx ; vr u; w x; ; z D E Eh L P(t)
stress wave speed elastic and plastic stress wave speeds, respectively wall thickness of the shell axial and radial velocities, respectively axial and radial displacements, respectively axial, circumferential and radial directions, respectively diameter of a shell Young’s modulus strain hardening modulus in true stress–true strain space length of a shell axial force acting on a shell
the three types of instability namely dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements), dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) and global bending (when the shell buckles in an Euler mode) develop according to the geometry of the shell, loading conditions and the material properties [11–15]. Thus, it is important to identify the factors causing the transition between the types of buckling and to determine the load causing a structural collapse. The principal dynamic e8ects on the elastic–plastic buckling behaviour of axially loaded cylindrical shells and many other thin walled structural elements are associated with the lateral inertia. The in/uence of axial inertia is neglected usually in theoretical studies. On other hand, the in/uence of the material properties on dynamic buckling is analysed mainly from the viewpoint of strain rate e8ects, which strengthen a material, but less attention is paid to the in/uence of the actual stress–strain relationship in the plastic range and to the approximation of this curve on the dynamic buckling predictions. A linear approximation of this relationship is assumed in some theoretical
Pmax R T0 Tc V0 x ;
xp ; p ; ep 0 x ; ; e
maximum load at t = 0 applied to the proximal end mean radius of a shell initial impact energy energy absorbed by a shell during the initial stable response initial velocity axial and circumferential total strains, respectively axial, circumferential and equivalent plastic strains, respectively material density Poisson’s ratio yield stress axial, circumferential and equivalent stresses, respectively
studies [4 –9,11,12,16,17], while non-linear approximations are considered usually in numerical analyses. In the particular case of an axially loaded cylindrical shell, sequential phases of compression and bending govern the deformation process, which form a complex loading=unloading path. Therefore, the phenomenon of dynamic buckling is very sensitive to the loading path in the plastic range, as observed in Ref. [4] for an idealised elastic–plastic model. From this viewpoint, it is important to analyse the in/uence of the material approximations on the buckling response. It appears, that the character of the initial stable phase of a shell deformation, when axial compression dominates the shell motion, in/uences the instability pattern, which develops later. Factors, which cause a transition from dynamic progressive to dynamic plastic buckling, have been identi0ed in [17–19] and they are closely associated with the material characteristics and the inertia properties of a shell. The purpose of this study is to explore the in/uence of approximations in the inelastic properties of a material on the predicted pattern of axisymmetric buckling and the in/uence of the axial inertia on the initiation and development of buckling of circular cylindrical shells subjected to a dynamic axial load.
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
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An axial impact on a stationary cylindrical shell is modelled as shown in Fig. 1 when varying the initial velocity of the striking mass between 10 and 75 m=s. All shells have a diameter D = 35 mm and a thickness h = 1:5 mm, and three lengths L = 3D; 4D or 5D. It is assumed that the shells are made of an aluminum alloy 6063-T5 [20] with elastic modulus E = 69 GPa, density = 2685 kg=m3 , yield stress 0 = 172 MPa and the actual static stress–strain relation in Fig. 2. The von Mises yield criterion is assumed with isotropic strain hardening. Material strain–rate e8ects are taken into account in some material models. Several multi-linear approximations of the actual stress–strain curve are considered in order to analyse their in/uence on the dynamic buckling phenomenon and these approximations are presented in the true stress–true strain space in Fig. 2. Two values for the hardening modulus, Eh , are used for the approximations 1 and 2 giving a strain energy equivalent to the
true stress, MPa
2. Shell geometry, material approximation and loading conditions 300
150
....... Approximation A1 - - - - Approximation A2 ____ Approximation A3
0 0.0
0.2
0.4
0.6
true strain Fig. 2. Material approximations. • • • Actual stress–strain relationship.
strain energy based on the actual stress–strain curve at 10% and 20% true strain, respectively. A hardening modulus of 880 MPa is assumed for the equivalent strain, ep , between 0 and 0.1, but Eh = 200 MPa for ep ∈ (0:1; 0:5) for approximation A1. A hardening modulus of 605 MPa is assumed for the equivalent strain, ep , between 0 and 0.2, but Eh = 156 MPa for ep ∈ (0:2; 0:5) for approximation A2. A piecewise linear presentation is used in approximation A3 when hardening moduli of 1072, 752, 448 and 250 MPa are associated with the true strain ranges of (0, 0.025), (0.025, 0.05), (0.05, 0.1) and (0.1, 0.5), respectively. Approximations 4 and 5 take into account material strain rate e8ects according to the Cowper–Symonds equation q e ep = D 0 − 1 (1) with D = 1; 288; 000 s−1 ; q = 4 [10] when using the corresponding static curves from approximations A2 and A3. In Eq. (1), 0 (ep ) is the static /ow stress, which depends on the plastic strain via isotropic hardening. 3. Modelling of the shell impact
Fig. 1. Shell geometry and the co-ordinate system.
A numerical simulation of the impact event was carried out using the FE code ABAQUS=Standard, version 5.8. It is assumed that an axisymmetric buckling mode develops in suNciently thick shells [12,13],
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so that 2D axisymmetric 8 nodes biquadratic solid elements are used in the simulation. The load is applied as a point mass attached to the nodes of a rigid body, which has an initial velocity V0 . All shells are modelled with three elements across the shell thickness while 60, 80 and 100 elements are used along the shells having L = 3D; 4D or 5D, respectively. The contact between the shell and the striker is de0ned using the ‘surface interaction’ concept together with a friction coeNcient 0.25 at the proximal end. The self-contact of the inner and the outer surfaces during the deformation of the shell are assumed frictionless. The axial and the radial degrees of freedom, u; w, of the nodes at the distal end are 0xed in order to model a clamped end, but no constraints are assumed for the degrees of freedom associated with the nodes at the proximal end when modelling a clamped-free shell. The modelled shells have no initial imperfections.
Fig. 3. Patterns of shell instability for a shell having L = 5D; V0 = 75 m=s; T0 = 2 kJ. (a) Dynamic progressive buckling—t = 0:356 ms, material approximation A3. (b) Dynamic plastic buckling—t = 0:324 ms, material approximation A2.
4. Patterns of buckling Several experimental studies on the axial impact of elastic–plastic cylindrical shells show that the response of a cylindrical shell, which buckles inelastically develops in two stages: initial instability pattern dominated by axial compression and a structural collapse when large deformations occur [20]. Recent studies reported in the literature [17–19] reveal that the initial instability pattern for elastic–plastic cylindrical shells subjected to an axial impact loading depends not only on the inertia characteristics of the shell, but is strongly in/uenced by the material hardening properties. Dynamic progressive buckling (Fig. 3(a)) dominates the shell response for a wide range of loading and material parameters [13–15], while dynamic plastic buckling (Fig. 3(b)) can develop for high velocity impacts and particular material properties [12,17–19]. It was observed in [17–19] for a particular cylindrical shell with a bilinear stress–strain approximation that the ratio between the material strain hardening and elastic moduli governs the initial pattern of instability. Dynamic plastic buckling is characterised by an initial wrinkling within a sustained plastic /ow along the shell length during the 0rst phase of deformation [12,17–19], while the strain localises in the case of progressive buckling [18,19].
A comparison between the initial shell instability and the 0nal buckling shapes of a shell having a length L = 4D and subjected to a 2 kJ impact applied with V0 =75 m=s is presented in Fig. 4 for di8erent material approximations. It is evident that the material approximations in/uence signi0cantly the buckling shapes. Dynamic plastic buckling characterises the initial instability pattern for a shell with the material properties approximation A2, while an initial instability in the dynamic progressive pattern is associated with the approximations A3 and A4. On other hand, the material approximations are more signi0cant when predicting the response to a high velocity impact, as shown in Fig. 5. It is evident that the buckling shapes di8er considerably for impact velocities higher than 30 m=s when using material approximations A1 and A3. The buckling patterns presented in Figs. 4 and 5 show that the approximations in the material properties together with the inertia e8ects in/uence the pattern of shell instability when assuming axisymmetric buckling. 5. Stress wave propagation The axial inertia of a shell plays an important role for higher impact velocity and has a signi0cant
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1227
Fig. 4. Initial shell instability and the 0nal buckling shapes of shells having L = 4D; V0 = 75 m=s; T0 = 2 kJ. (a) Initial, (b) approximation A2, t = 0:235 ms, (c) approximation A2, t = 2:83 ms, (d) approximation A3, t = 0:235 ms, (e) approximation A3, t = 2:997 ms, (f) approximation A4, t = 0:235 ms, (g) approximation A4, t = 2:540 ms.
e8ect on the buckling initiation patterns. A high velocity impact causes an instantaneously applied force on the shell and initiates elastic and plastic stress waves, which propagate along the shell. It was shown in [17,18] that the assumption of a biaxial stress state, (x = 0; = 0), can be used with suNcient accuracy to analyse the transient deformation process in circular cylindrical shells. A plastic wave can propagate at di8erent speeds in shells depending on the assumed material model. The governing equations of motion of an elastic–plastic medium in a biaxial stress state (x = 0; = 0) are [18]: At wt + Ax wx + b = 0; where
0 At = 0 0
0 0
0 0 0 ; R 0 0
0 0 b= ; − −w=R ˙
(2)
0 −1 Ax = 0 0
−1 0 0 0
0 0 0 0
˙ ] T ˙ x ; w; w = [ u;
0 0 ; 0 0
and were used to obtain the speeds of the elastic and plastic waves, which can propagate in axially loaded cylindrical shells. The plastic stress waves in shells made of a strain rate sensitive material have a constant speed c = ±(E=(1 − 2 ))1=2 = cp ;
(3)
when the strain–rate e8ects are taken into account according to the Cowper–Symonds equation [19]. The stress wave speeds for strain rate insensitive materials depend on the stress state in the plastic range and e.g. for a von Mises material the wave speeds can be expressed as [18,19]
1=2 c=± ; (4) ( − 2 ) where (2x − )2 1 ; + g (e ) = E (2e )2 (2x − )(2 − x ) = − + g (e ) ; E (2e )2 1 (2 − x )2 = + g (e ) E (2e )2
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D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238 σθ/σ0 p
2
e
c /c
1
Yield locus
0 0.1 -1
-λ = 0.01
−λ = 1 (elastic)
-2
-2
-1
0
1
2 σx/σ0
Fig. 6. Stress wave speeds in isotropic elastic–plastic materials with a linear strain hardening and the von Mises yield criterion ( = 1; 0:1; 0:01).
Fig. 5. Final buckling shapes of shells having L = 3D subjected an impact with T0 = 1:5 kJ for material approximation A1 (b–e) and material approximation A3 (g–j); (b,g) V0 = 10 m=s, (c,h) V0 = 30 m=s, (d,i) V0 = 60 m=s, (e,j) V0 = 75 m=s.
ep
and where = g(e ) is the stress–strain relationship in the true stress–true strain space with g (e ) = (1 − )=E and = Eh =E for a material with linear strain hardening, Eh . An illustration of the stress wave speed dependence on the stress state (x ; ) and for di8erent hardening ratios , are presented in Fig. 6 as a unit vector (cp =ce ) perpendicular to the initial yield locus for the von Mises yield condition ( varies between 0.0036 and 0.0115 for the approximations of Al 6063-T5 in Section 2). It is evident that the elastic wave speed is an upper bound for plastic wave speeds in a strain rate insensitive material. The minimum values for the plastic wave speeds show a strong dependence on the hardening modulus, which can be compared to the corresponding proportionality to the hardening
modulus in the uniaxial case. It is shown in [18], however, that the minimum values for the plastic wave speed in strain rate insensitive materials are always larger than the plastic wave speed in a uniaxial medium as p p p cmin; Mises ¿ cmin; Tresca ¿ cuniaxial ;
(5)
p p where cmin; Mises and cmin; Tresca are the minimum values for the plastic wave speeds, which propagate in shells made of materials obeying the von Mises or Tresca yield criteria. For a bilinear elastic–plastic material the corresponding expressions for the plastic wave speeds in circular shells are obtained as [17–19] p 1=2 2 cmin; Mises = ±(E=) {4 [4 (1 − )
+3(1 − )]−1 }1=2
(6)
for ( =x =2; x ¡ 0; ¡ 0) and (( =x =2; x ¿ 0; ¿ 0), p cmin;
Tresca
= ±(E=)1=2 {2 [2 (1 − 2 ) √ +(1 − ) 3]−1 }1=2
(7)
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
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Fig. 7. Development of the buckling shape for a shell with L = 4D; material approximation A2, V0 = 75 m=s, T0 = 2 kJ; (a) initial, (b) t = 0:217 ms, (c) t = 0:557 ms, (d) t = 0:735 ms, (e) t = 0:951 ms, (f) t = 1:176 ms, (g) t = 1:83 ms, (h) t = 2:83 ms.
Fig. 8. Development of the buckling shape for a shell with L = 4D; material approximation A3, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:233 ms, (c) t = 0:638 ms, (d) t = 0:963 ms, (e) t = 1:2 ms, (f) t = 2:54 ms, (g) t = 2:977 ms.
for ( = x ; x ¡ 0; ¡ 0) and ( = x ; x ¿ 0, ¿ 0), while the uniaxial plastic wave speed is p cuniaxial = ±( E=)1=2 :
(8)
The development of the buckling shape with time of a shell having a length L = 4D subjected to a
75 m=s axial impact is presented in Figs. 7–10 for the material approximations A2, A3, A4 and A5, respectively. It is evident that di8erent buckling patterns are associated with the various approximations. The axial inertia of the shell plays a signi0cant role for such a high velocity impact so that the transient deformation process becomes an
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Fig. 9. Development of the buckling shape for a shell with L = 4D; material approximation A4, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:235 ms, (c) t = 0:480 ms, (d) t = 0:732 ms, (e) t = 1:018 ms, (f) t = 2:54 ms.
Fig. 10. Development of the buckling shape for a shell with L = 4D; material approximation A5, V0 = 75 m=s; T0 = 2 kJ; (a) initial, (b) t = 0:235 ms, (c) t = 0:340 ms, (d) t = 0:792 ms, (e) t = 1:66 ms, (f) t = 2:243 ms.
important part of the response. Figs. 11(a) – (c) present the axial and the circumferential strains along the shell mid-surface at several times for three of the material models. The axial strain distribution at t = 0:217 ms in Fig. 11(a) (approximation A2) shows that the shell has buckled with wrinkles along the
entire length (dynamic plastic buckling, Fig. 7(b)). The particular impact velocity of 75 m=s causes instantaneous plastic deformations at the proximal end of the shell and a plastic wave having a speed of approximately 550 m=s. The primary plastic wave and the secondary one resulting from the
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
strains 0.1
εθ
0.235 0.217
0.0
0.144 0.055 msec 0.144
0.217
-0.1
0.100
0.235
-0.2
(a)
εx
0.000
0.035
0.070
0.105
0.140
x, m
strains εθ
0.1 0.0
0.074 msec
/ 0.181
0.215
-0.1
/ 0.154
/ 0.107
-0.2
(b)
0.000
εx
0.035
0.070
0.105
0.140
x, m strains 0.1
εθ
0.235
0.0
0.050 msec 0.190
0.100
0.07
-0.1 0.235
-0.2
0.000 (c)
εx
0.035
0.070
0.105
0.140
x, m
Fig. 11. Strain distribution until the initiation of buckling in a shell having L = 4D and di8erent material approximations; V0 = 75 m=s; T0 = 2 kJ. (a) material approximation A2, (b) material approximation A3, (c) material approximation A4.
re/ection of the elastic wave from the distal end meet at t ≈ 0:140 ms when the entire length of the shell is under large axial compression and until
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t ≈ 0:190 ms buckling develops within a sustained plastic /ow. High velocity impacts on a shell made of an elastic– plastic material with approximation A3 cause plastic waves to propagate at di8erent speeds with the largest strains travelling slowest [21]. The strain patterns in Fig. 11(b) (approximation A3) show that a 75 m=s velocity impact causes instantaneous axial compressive strains of approximately 0.12, which is associated with Eh = 250 MPa. The plastic wave emanating from the proximal end and having a lower speed (approximately 350 m=s) can propagate only a short distance (≈ 24 mm) before the radial displacements near this particular end start to grow rapidly (Fig. 8(b)) for t ¿ 0:07 ms since the radial inertia cannot any longer support the unbuckled shape. The elastic unloading across the shell thickness interrupts the further propagation of the plastic stress wave so that a localisation of strains occur near the proximal end of the shell causing a further progressive folding, although small strains at higher speeds continue to propagate. The axial strains along the shell with material approximation A4, which includes material strain rate e8ects, are presented in Fig. 11(c). In this case, the plastic waves propagate at the elastic wave speed and the axial strains, although they vary in magnitude along the shell, increase with time along the entire length. Similarly to approximation A2 in Fig. 11(a), buckling occurs within a sustained plastic /ow, however even when the stresses start to decrease for the visco-plastic material, the strains can continue to increase [19]. This material behaviour causes a strain localisation near to the shell ends (Fig. 11(b)) and further progressive buckling. It is evident for material approximation A5 in Fig. 10(b) that the strain localisation at both ends of a shell is even more signi0cant than for material approximation A4. This phenomenon occurs because the static stress–strain curve for approximation A5 has lower strain hardening properties than material approximation A4. The development of the buckling shapes presented in Figs. 7–10 reveals that regardless of the initial pattern of the shell instability, dynamic buckling always continues progressively provided suNcient external energy is available. The initiation of buckling, however, a8ects the 0nal buckling shapes as shown in Fig. 4.
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6. Axial inertia e&ects The responses of shells having di8erent lengths are examined in order to explore further the in/uence of the axial inertia on the initial instability pattern and the 0nal buckling shape. The 0nal buckling shapes of shells having L = 5D; 4D and 3D and subjected to 2 kJ impact with an initial velocity 75 m=s are presented in Fig. 12 for material approximations A3 (Fig. 12(b),(f),(j)), A2 (Fig. 12(c),(g),(k)) and A1 (Fig. 12(d), (h),(l)). It is evident that the number of the developed folds depends on the material approximation but not on the shell length. However, the location of the folds depends on the shell length for shells made of material A3. An explanation for this phenomenon is related to the axial inertia e8ects on the buckling mechanism. Material approximations A1 and A2 allow larger strains to propagate at higher speeds along the shell due to the larger hardening moduli (see Fig. 11(a)). For these material approximations, the plastic waves can propagate along the shell lengths (L=5D; 4D and 3D) causing dynamic plastic buckling without initial strain localisation near to the proximal end. A localisation of strains develops near to the distal end due to the re/ected stress wave causing further progressive folding near this particular end (see Fig. 7). Plastic stress waves for material approximation A3 propagate at di8erent speeds from the proximal end. The slower stress waves carry larger stresses and strains thus causing strains with di8erent magnitudes to propagate along the shell. In this case, the strain localisation near to the proximal end causes progressive folding to develop initially near this particular end as shown in Figs. 8 (b),(c) (L = 4D) and 13 (b),(c) (L = 3D). However, the re/ected stress wave from the distal shell end of the shell having L = 3D initiates the strain localisation and a rapid growth of the radial displacements more quickly due to the shorter length as evident when comparing Figs. 8 and 13. Two complete wrinkles near to the distal end characterise the 0nal buckling shape of the shell with L = 3D, while only one complete wrinkle develops in the shell having L = 4D and no complete wrinkle is observed for the shell of L=5D in Fig. 12(b). The piecewise linear material approximation can have an additional e8ect in connection with the rate
of loading. The development of the buckling shapes of a shell with L = 4D and a material approximation A3, which is subjected to a 1:5 kJ impact is presented in Figs. 14 and 15 for V0 = 60 and 30 m=s, respectively. The higher velocity impact causes larger plastic strains to develop near to the proximal end at the early stage of the shell response and these strains propagate relatively slowly. The radial inertia cannot support such high stresses and the shell starts to buckle near to the proximal end as evident in Fig. 14 (b),(c). Smaller plastic strains, which propagate at a higher speed, emanate from the proximal end as a result of a 30 m=s impact. These strains are too small to cause any initial buckling near to the proximal end, but the re/ected stress wave produces buckling near to the distal end, as shown in Fig. 15(b). One wrinkle develops near the proximal end at a much later time (Fig. 15(d)) and further progressive buckling forms the 0nal buckling shape. A 60 m=s impact also causes progressive buckling, however, the sequence of the wrinkle development is di8erent, as evident in Fig. 14. 7. Buckling load for high velocity impacts Unlike the elastic and elastic–plastic response to a low velocity impact, when the load increases from zero and buckling occurs upon reaching a critical value, the maximum load on a shell resulting from a high velocity axial impact occurs instantaneously and the associated strains exceed the elastic limit of the material. The peak load at t = 0 results from the axial stress discontinuity produced at t = 0 and is independent of the striking mass [19]. As a high velocity impact is characterised by a signi0cant radial inertia of the shell, the peak load may or may not cause buckling depending on the initial kinetic energy of the striker [19]. A small initial kinetic energy applied with a high impact velocity can be absorbed only by an axial compression and a uniform expansion of the tube, but larger energies can cause initial instability and a further collapse [18]. A relationship between the impact velocity and the maximum value of the applied load, which occurs at t = 0, can be obtained when analysing the stress wave propagation in an elastic–plastic shell [18,19]. The initial response of an elastic–plastic cylindrical shell to
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
1233
Fig. 12. Final buckling shapes of shells subjected to an impact with T0 = 2 kJ; V0 = 75 m=s (a–d) L = 5D, (e–h) L = 4D, (i–l) L = 3D, (b,f,j) approximation A3, (c,g,k) approximation A2, (d,h,l) approximation A1.
an axial impact is characterised by large axial compressive strains and negligible circumferential strains [18], which leads to ≈ x =2; x ¡ 0; ¡ 0 when using the von Mises yield criterion and an initiation p of plastic waves having speeds cp ≈ cmin . In the case of a piecewise linear stress–strain relationship (approximation A3), the instantaneous plastic strains depend on the impact velocity and they are
associated with the corresponding hardening modulus for di8erent parts of the stress–strain curve. Similarly to the case of a strain rate insensitive isotropic material with a linear hardening [18,19], the peak load, which acts on a shell made of an elastic–plastic material with a piecewise linear material approximation, is a function of the material properties and the impact velocity.
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Fig. 13. Development of the buckling shape for a shell with L = 3D; material approximation A3; T0 = 2 kJ; V0 = 75 m=s; (a) initial, (b) t = 0:302 ms, (c) t = 0:546 ms, (d) t = 0:751 ms, (e) t = 1:328 ms, (f) t = 2:323 ms, (g) t = 3:55 ms.
Fig. 14. Development of the buckling shape for a shell with L = 4D; material approximation A3; T0 = 1:5 kJ; V0 = 60 m=s; (a) initial, (b) t = 0:322 ms, (c) t = 0:504 ms, (d) t = 0:918 ms, (e) t = 1:598 ms, (f) t = 2:612 ms.
The di8erential relations along the characteristics of the problem determined by Eq. (2) lead to [18] dvx vr dx dx ∓ cp = − (cp )2 along = ±cp dt dt R dt (9) and vr d
dx = ; + dt dt R
(10a)
dx dvr = along = 0; (10b) dt dt which results in equation
2 (cp )2 d x + (cp )2 d = ±cp dvx : 1+ (11)
R
Using the initial condition vr = 0 at x = L;
(12)
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1235
the stress discontinuity occurs in the nth linear portion of the stress–strain diagram having a hardening modulus (Eh )n , an expression for the axial stress, x|x=L , at t = 0 can be obtained as x|x=L n−1 p = %(x )i + (cmin )n
i=1
n−1 %(x )i V0 − p (cmin )i i=1
: (17)
p The terms %(x )i =(cmin )i in Eq. (17) can be denoted formally as velocity increments, (Tvx)i , so that the peak load, which can result from an axial impact on a shell made of a piecewise linear material (e.g. approximation A3), can be expressed as √ p Pmax; Mises =2&Rh = 20 = 3 + (cmin )n
Fig. 15. Development of the buckling shape for a shell with L=4D; material approximation A3; T0 = 1:5 kJ; V0 = 30 m=s; (a) initial, (b) t=0:765 ms, (c) t=2:506 ms, (d) t=3:762 ms, (e) t=5:672 ms.
the dynamic discontinuity condition on the wave front is obtained as p (x ; )vx = 0; dx − dcmin
i.e. the velocity on the wave front is
1 x d Sx vx = p : 0 cmin The above relationship can be presented as n n 1 (x )i+1 d Sx 1 %(x )i V0 = = p p (cmin )i (cmin )i (x )i i=1 i=1
(13)
(14)
(15)
for a piecewise linear material and initial condition p vx|x=L = V0 and where (cmin )i is the minimum plastic wave speed associated with the linear strain hardening modulus (Eh ). The piecewise linear axial stress, (x )i , can be expressed in the form √ (16) (x )i = 2(e )i = 3; where (e )i is the equivalent stress associated with the particular linear region of the stress–strain relation and when assuming ( )i = (x )i =2. If the impact velocity is high enough to cause instantaneous plastic deformations at the proximal end of the shell, then the peak load, Pmax; Mises , can be related to the stress discontinuity at t = 0. Assuming that the magnitude of
× V0 −
n−1
(Tvx )i
+
i=1
which reduces to [18] √ p x|x=L = 20 = 3 + cmin V0
n−1 i=1
p (cmin )i (Tvx )i
; (18a) (18b)
for a shell made of an elastic–plastic strain rate insensitive material with a single hardening modulus. The predictions for peak loads of 50.9 and 49:7 kN resulting from a 75 m=s impact on a shell with L = 4D when assuming the material approximations A2 and A3 (Eq. (17)) compare well with the corresponding numerical values of 50.8 and 49:2 kN in Figs. 16(a) and (b). Because of the similar magnitudes of the stress discontinuity at t = 0, the initial deformation phases for both material approximations are identical. Both force–time and force–displacement histories in Fig. 16 are characterised by a rapid decrease of the suddenly applied force. The associated axial compression of the shell is about 3 mm without any bending deformations. This phase lasts for about 0:035 ms and during this time the plastic waves travel at ∼550 m=s and propagate a distance of 20 mm, which is comparable to the critical wavelength of the shell (12:75 mm [22]). Thus, buckling develops within a sustained plastic /ow near to the proximal end of the shell. After this time the shell responses develop di8erently due to the di8erent hardening
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P, N 40000
20000
0 0.000
0.002
0.001
(a)
0.003
t, sec
P, N 40000
20000
0 0.000
(b)
0.002
0.001
0.003
t, sec P, N 40000
20000
0
0 0.00 (c)
length of the shell with the material approximation A2 due to the higher speed of the plastic wave travelling along the shell (Fig. 7(b)). In contrast, the lower plastic wave speeds associated with the larger strains for the material approximation A3 cause a further localisation of the strains near to the proximal end of the shell and local bending occurs (Fig. 8(b)). The force–time histories in Figs. 16(a) and (b) indicate that a shell made of a material approximated as A2 can support a shape with only small radial displacements up to t ≈ 0:24 ms and a rapid buckling occurs thereafter (Fig. 7). For t ¡ 0:24 ms, an almost constant force acts on the proximal end of the shell. By way of contrast, a shell with material approximation A3 responds by a sudden buckling (Fig. 8), which is manifested as a continuous decrease of the force acting on the impacted end. It is evident, that the initial stable response of a shell, which is characterised by an axial compression, lasts longer for shells, which are made of inelastic materials with high strain hardening properties. This behaviour is due to the plastic waves, which propagate at higher speeds in these materials and cause large axial compressive strains to develop along the entire length of a shell before any elastic unloading occurs across the shell thickness. Fig. 16(c) shows that a shell made from material A2 (solid curve) can compress signi0cantly (u|x=L ≈ 0:017 m) before the development of large radial displacements. In this case, the stress discontinuity at t = 0 can be used to estimate the upper value for the impact energy, Tc , which the shell can absorb during the initial stable compression phase as [10]
e0 e de ≈ 2&RhLe0 (0 + e0 )=2; (19) Tc = 2&RhL
0.02
0.04
0.06
0.08
crushing distance, m
Fig. 16. (a) Force–time history at proximal end for initial velocities 75 m=s; L=4D; material approximation A2. (b) Force–time history at proximal end for initial velocities 75 m=s; L = 4D; material approximation A3. (c) Force–displacement histories at proximal end for initial velocity 75 m=s (T0 = 2 kJ); L = 4D; —— material approximation A2, - - - - - - material approximation A3.
characteristics associated with the material approximations A2 and A3. Small wrinkles within a sustained plastic /ow continue to develop along the
when neglecting elastic e8ects and where e0 and e0 are the equivalent stress and the corresponding equivalent plastic strain at t = 0. The absorbed energy estimated by Eq. (19) is an upper bound for the energy absorbed during the initial compression phase, since the stress along the tube is smaller than the stress discontinuity at t = 0 at the proximal shell end. However, if it is assumed that the axial stress along the tube is equal to the stress discontinuity determined by, e.g. Eq. (18), then the equivalent plastic strain can be obtained as e0 = ep = (e0 − 0 )=Eh :
(20)
D. Karagiozova, N. Jones / International Journal of Non-Linear Mechanics 37 (2002) 1223 – 1238
For example, a shell having a length L = 4D and made of a material approximated as A2 can absorb up to 0:8 kJ in axial compression when impacted at 75 m=s according to Eqs. (18) – (20) and (6). It can be expected that energies ¡ 0:8 kJ (corresponding to an impact mass of 0:284 kg for the initial velocity of 75 m=s) cannot cause an inelastic collapse of the shell, but only an axial compression with small wrinkles along the shell developed within a sustained plastic /ow, which do not cause a local shell instability. For example, the deformed shape of the shell, presented in Fig. 4(b) at t = 0:235 ms is associated with an absorbed energy of about 0:6 kJ at this particular time. Consequently, the maximum value of the dynamically applied force to a shell, which buckles inelastically, does not play the same role as the critical dynamic load for elastic shells because of the axial inertia e8ects and the energy dissipated by the inelastic material behaviour during the initial stable response of the shell. While the maximum dynamic load is critical for the elastic shells and depends only on the radial inertia of the shell, the maximum dynamic load applied to an inelastic shell can be considered as a critical buckling parameter only in relation to the applied impact energy and the proportion of this energy, which can be absorbed during the initial axial compression phase. 8. Conclusions The in/uence of dynamic e8ects in the elastic– plastic buckling phenomenon is considered from the viewpoint of a transient process in cylindrical shells subjected to an axial impact. It is shown that the initiation of buckling for a high velocity impact is sensitive to the material properties, particularly to the hardening modulus, and the plastic wave speed determined by this modulus can lead either to dynamic plastic buckling or to dynamic progressive buckling. The material approximation can change the predicted pattern for the initial instability and the 0nal buckling shape of a shell. A regular buckling shape can occur for strain rate insensitive materials when buckling develops within a sustained axial plastic /ow, while the localisation of strains leads to progressive buckling in shells made of materials having a low strain hardening behaviour or strain rate sensitive properties. The
1237
rate of loading causes an additional e8ect on the buckling shapes of the shells made from an elastic–plastic material with piecewise linear strain-hardening characteristics because the stresses then propagate at different wave speeds along a shell. It is shown that the local inelastic instability of a shell (a structural collapse) subjected to a high velocity impact depends not only on the magnitude of the applied load, but also on the proportion of the initial kinetic energy, which can be absorbed by the axial compression of a shell during the initial stable response.
Acknowledgements The work outlined in this paper was supported by EPSRC Grant GR=M=750044.
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[11] W. Abramowicz, N. Jones, Transition from initial global bending to progressive buckling of tubes loaded statically and dynamically, Int. J. Impact Eng. 19 (1997) 415–437. [12] A.L. Florence, J.N. Goodier, Dynamic plastic buckling of cylindrical shells in sustained plastic /ow, J. Appl. Mech. 35 (1968) 80–86. [13] K. Murase, N. Jones, The variation of modes in the dynamic axial plastic buckling of circular tubes, in: N.K. Gupta (Ed.), Plasticity and Impact Mechanics, Wiley Eastern Limited, New Delhi, 1993, pp. 222–237. [14] Li Ming, Wang Ren, Han Mingbao, An experimental investigation of the dynamic axial buckling of cylindrical shells using a Kolsky bar, Acta Mech. Sinica 10 (1994) 260–266. [15] T. Kurokawa, T. Sasaki, H. Suzuki, Y. Iamada, Axisymmetric collapse of circular tubes due to longitudinal impact, Proceedings of the Sino-Japanese Symposium on Deformation=Fracture of Solids, Huangshan= China, 1997, pp. 9 –16.
[16] D. Karagiozova, N. Jones, Dynamic elastic–plastic buckling phenomena in a rod due to axial impact, Int. J. Impact Eng. 18 (1996) 919–947. [17] D. Karagiozova, N. Jones, Dynamic elastic–plastic buckling of circular cylindrical shells under axial impact, Int. J. Solids and Struct. 37 (2000) 2005–2034. [18] D. Karagiozova, M. Alves, N. Jones, Inertia e8ects in axisymmetrically deformed cylindrical shells under axial impact, Int. J. Impact Eng. 24 (2000) 1083–1115. [19] D. Karagiozova, N. Jones, In/uence of stress waves on the dynamic plastic and dynamic progressive buckling of cylindrical shells under axial impact, Int. J. Solids Struct. 38 (2001) 6723–6749. [20] H.E. Lindberg, A.L. Florence, Dynamic Pulse Buckling, Martinus Nijho8, Netherlands, 1987. [21] W.K. Nowacki, Stress Waves in Non-Elastic Solids, Pergamon Press, UK, 1978. [22] W. Abramowicz, N. Jones, Dynamic axial crushing of circular tubes, Int. J. Impact Eng. 2 (1984) 263–281.