COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Elastoplastic Impact of the Sphere upon the Uflyand-Mindlin Plate
I. A. Lokteva 1*, A. A. Loktev 2
1
2
Department of Chemistry, Voronezh State University, Voronezh, 394006 Russia
OOO “Konmark”, Voronezh, 394000 Russia
Email: [email protected]
Abstract The elastoplastic models of transverse impact of a cylindrical indenter on an elastic isotropic UflyandMindlin plate are investigated. The displacements of the points of the plate outside of the contact area occur because
the wave of transverse shear travels at final velocity. This wave appears at the moment of the impact and represents the
surface of strong discontinuity. Behind the front of the transverse wave the values, which it is necessary to find, are
represented as one-term ray expansions. The local shear of the material happens in a quasistatic way; its dependence on
the contact force is defined by the elastoplastic dependence. The influences of the elastoplastic qualities of the plate
material at the contact point and the inertia of the contact disc on the contact force are studied.
Key words: transverse impact, Uflyand-Mindlin plate, one-term ray expansions, contact force
INTRODUCTION
The problems of the dynamic contact of the indenter and the plate were considered in papers [1-3]. In paper [1] to
describe the function of the curves the row expansion according to Bessel’s equations was used. In papers [1-3] to
determine the values behind the wave fronts the ray rows are used, which represent the sedate time rows with variable
coefficients, which depend on the parameters of the indenter and the target. As far as the time of the contact is short, we
can circumscribe with the zero term of the row, which was seen in papers [2, 3]. In papers [1-3] to determine the local
shear the Hertz’s theory of the elastic interaction was used, but as the experimental and theoretical results show [5], if
the initial velocity of the impact exceeds a certain value, then the elastoplastic qualities of the interacting bodies
substantially influence the dynamic characteristics.
Basing on the Uflyand-Mindlin wave equations and the ray method the influence of elastoplastic qualities of different
modules of the local shear on the contact force and the dynamic curve is investigated in the paper. Supposedly the plate
is rather long and the reflected wave does not have time to reach the boundaries of the contact area before the process
of interaction is completed.
PROBLEM FORMULATION
The indenter with the mass of m is moving perpendicularly to the plate and strikes it at the speed of V0 (Fig.1a). At the
moment of the impact in the plate forms the contact area with the radius r, which depends on the radius of the indenter
and the value of the contact shear a.
The moving of the indenter after the beginning of the impact is described by the following equation:
&& ) = − P ( t ) .
m (α&& + w
(1)
The equation of the contact area movements:
&& = − 2π rQr
ρ hπ r 2 w
r = r1
+ P (t ) .
(2)
The process of the impact interaction occurs if the following initial conditions are realized:
α& t =0 =V , w& t =0 = 0 .
(3)
⎯ 1141 ⎯
(а) before impact
(b) after impact
(c) Low view
Figure 1: Scheme of the shock interaction of a rigid impactor and a plate
The dynamic behavior of the elastic isotropic plate in the polar coordinates system r, ϕ is described by UflyandMindlin equations, which take into consideration the rotary inertia and the deformation of the transverse shear [3]:
∂M r
1
ρ h3 &&
+ Qr =
βr ,
M r − Mϕ +
∂r
12
r
(4)
∂Qr Qr
+
= ρ hW& ,
∂r
r
(5)
⎛ ∂β&
⎛ β&
β& ⎞
∂β& ⎞
M& r = D ⎜ r + σ r ⎟ , M& ϕ = D ⎜ r + σ r ⎟ ,
r ⎠
∂r ⎠
⎝ ∂r
⎝ r
(6)
⎛ ∂W & ⎞
− βr ⎟ ,
Q& r = K μ h ⎜
⎝ ∂r
⎠
(7)
(
)
where r and ϕ are the polar radius and angle, M r and M ϕ - bending moments, Qr - the transverse force, β&r - the
& - the velocity of the sag, ρ - the
angle velocity of the normal to the median surface of the plate in r direction, W = w
(
density , K = π 2 12 , μ - the module of the shear, σ - the Poisson’s ratio, D = E 1−σ 2
)
−1 3
h /12, t –time, the point above
the values shows the time-derivative.
METHOD OF SOLUTION
After the impact the transverse wave begins to propagate from the contact area in the plate, the front of this wave
representing the cylindrical surface of strong discontinuity, expanding at the speed of G.
⎯ 1142 ⎯
Outside the contact area behind the front of the wave surface the solution is seen as the ray row by the coordinate and
time [2]:
k
∞
⎛ r − r0 ⎞
⎛ r − r0 ⎞
1
Z(r,t ) = ∑ ⎡⎣Z,( k ) ⎤⎦
t−
H ⎜t −
⎜
⎟
⎟⎟,
⎜
⎟
⎜
t =r / G
k!
G ⎠
G ⎠
⎝
⎝
k =0
(8)
where [Z,(k)]=Z +,(k) − Z −,(k)=[∂ kZ/∂ t k] are the leaps of the derivatives of k-degree by the time t from the equation Z on
the wave surface ∑, i.e. if t=(r−r0)/G(α), r0 – the initial radius, indexes “+” and “–” mean that the value is found directly
in front of and behind the wave front respectively, H(t) – the one-term Heviside’s function.
Considering only the first term of the ray row (8) from the equations (4-7) in the paper [3] the velocity of the
displacement wave is determined as well as the dynamic condition of the mutuality:
G=
Kμ
ρ
, Qr = − ρ GhW .
(9)
Adding Qr and W into Eqs. (1) and (2) and considering that for the spherical end of the impactor r12 = Rα , we
receive the non-linear integral-differential equation relatively α and Р:
⎤ P (t )
2G 1 2 ⎡ 1 t
⎛ P && ⎞
−α ⎟ = −
− ∫ P ( t ) dt − α& ⎥ +
α
.
⎢
⎝ m
⎠
R1 2
⎢⎣ m 0
⎥⎦ ρ hπ R
α ⎜−
(10)
The local shear is connected with the contact force by the following ratios [6,7], received from solving the contact
problems:
(1) Hertz’s elastic model:
α = bP 2 3 ,
(11)
(2) Elastoplastic model [6]:
⎧ bP 2 3 ,
dP dt > 0, Pmax < P1 ,
⎪⎪
α = ⎨(1 + β ) c1 + (1 − β ) Pd , dP dt > 0, Pmax > P1 ,
⎪
23
dP dt < 0, Pmax > P1 ,
⎪⎩ b f P + α p ( Pmax ) ,
(12)
3) Elastoplastic model [7]:
⎧bP 2 3 ,
dP dt > 0, P < Pb ,
⎪⎪
23
α = ⎨bP + Pd ,
dP dt > 0, P > Pb ,
⎪ 23
dP dt < 0, Pmax > Pb .
⎪⎩bP + Pmax d ,
((
where b = 9π 2 (k1 + k )
2
) 16R )
13
(
(13)
)
(
)
, k1 = 1 − σ 12 E1 , k = 1 − σ 2 E , χ = πmλ , P1 = χ 3 ( 3R ( k1 + k ) 4 ) , m is
2
minimal plastic constant of the interacting bodies, λ = 5.7, b f = R −f 1 3 (3(k1 + k ) 4 )
23
, R p−1 = R −1 − R −f 1 ,
1 2 −3 2
R f = (4 3(k1 + k ))Pmax
χ
, α p (P max ) = (1 − β )P max (2 χR p )−1 , с1 = 3χ 1 2 (k1 + k ) 8 , β = 0.33 , d = 1 2 χR , σ1, E1 –
Poisson ratio and modulus of intender respectively.
After we add the expressions (11)-(13) into the formula (10) we receive the integral-differential equations relatively
the contact force. The solution of these equations is found numerically with the help of a computer on each interval
(n −1)τ ≤ t ≤ nτ , supposing that within the single interval the contact force changes in a linear way:
P& ( nτ ) = ( Pn − Pn −1 ) τ ,
(14)
where τ is a step of integration.
The solution of non-linear equations is brought as the graphics of dependence P(t).
⎯ 1143 ⎯
NUMERICAL ANALYSIS
For the illustration of the achieved results we will consider the numerical example and investigate the dependence of
the contact force on the model of the contact interaction and the elastoplastic qualities of the plate. The parameters of
the studied construction take the following values: m = 0.3 kg, V0 = 10 m/s, h = 100 mm, E1 = E = 200 GPa,
σ 1 = σ = 0.3 , ρ = 7850 kg/m 3 .
In fig.2 we can see the dependence of the contact force on the time, the curves 1, 2 and 3 are found with the use of ratios
(11), (12) and (13) respectively and the curve 4 is taken from [5].
From fig.2. it is seen that the model (12) combined with wave approach gives the best approximation to the experiment
results and under the rates ≈ 10 m/s and more it is necessary to consider the elastoplastic properties of the striking
bodies in analysis.
Figure 2: Dependence of contact force on time
REFERENCES
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102-109 [The transverse elastic strike against the round plate. Mechanics of a solid body, 1971; 6: 102-109].
2. Yu Rossikhin A, Shitikova MV. A ray method of solving problems connected with a shock interaction. Acta
Mechanica, 1994; 102: 103-121.
3. Yu Rossikhin A, Shitikova MV. Impact of an elastic sphere upon a Timoshenko beam an Uflyand-Mindlin plate
with account for middle surface extension. Izvestija Vuzov. Stroitelstvo, 1996; 6: 28-34 (in Russian).
4. Loktev AA. Elastic transverse impact on an orthotropic plate. Technical Physics Letters, 2005; 31(9): 767-769.
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⎯ 1144 ⎯