# A Green’s Function Approach for Dynamic Stress Analysis of Spherical Shell under the Isotropic Impact Load

Authors: Jincheng Lv; Shike Zhang; Xinsheng Yuan
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A Green’s function approach is developed for the analytic solution of thick-walled spherical shell under an isotropic impact load, which involves building Green’s function of this problem by using the appropriate boundary conditions of thick-walled spherical shell. This method can be used to analyze displacement distribution and dynamic stress distribution of the thick-walled spherical shell. The advantages of this method are able(
1
) to avoid the superposition process of quasi-static solution and free vibration solution during decomposition of dynamic general solution of dynamics, (
2
) to well adapt for various initial conditions, and (
3
) to conveniently analyze the dynamic stress distribution using numerical calculation. Finally, a special case is performed to verify that the proposed Green’s function method is able to accurately analyze the dynamic stress distribution of thick-walled spherical shell under an isotropic impact load.


---

## Body

## 1. Introduction

In practical engineering, isotropic spherical surfaces or shells have a wide range of uses in thermodynamics analysis, electromagnetic analysis, tectonic dynamics analysis of the earth, and so forth. Since Hu had the pioneering research for the general theory of elasticity of isotropic spherical medium [1], more and more researchers attached importance to the study of isotropic spherical surface problems, and they have made great developments. Isotropic spherical surface or shell problems are investigated mainly through the analytic solution method, test method, and numerical computation method in [2–6].From mathematics and computational perspective, displacement and stress distribution of isotropic spherical surfaces or shells under the isotropic impact load can be obtained by the method of characteristics, integral transformation method, generalized ray method, approximation method, and domain decomposition method in [7–11]. In the 1990s, tectonic dynamics of the earth was considered as a new subject, and researchers have studied deformation problems of isotropic spherical surfaces and shells with multilayer hybrid model. Here Ding et al. in [12] introduced three displacement functions to motion equations of elasticity problems in spherically isotropic media; this effectively simplifies motion equations and improves the solution method of ordinary differential equations for dynamics problems. An analytic solution concerning elastic spherical media under the impact load was given by using characteristic function expansion method to decompose dynamics into quasi-static elastic solution and free vibration solution at inhomogeneous boundary condition in [13, 14]. These methods, however, are generally suitable for analytical solutions in simple spherical media, which could lead to difficulties in analyzing displacement and dynamic stress distribution of thick-walled spherical shell under the isotropic impact load, especially when using numerical computation to analyze these problems.To overcome difficulties mentioned above, Green’s function of elastic solid sphere was established to solve spherical surface and shell problems under an isotropic impact load in [15–17]. However, the established analytic solutions and mathematical expression of dynamic stress distribution are not well suitable for the thick-walled spherical shells.The objective of this paper is to develop a Green function of the thick-walled spherical shells under an isotropic impact load of solid sphere based on the research ideas of the aforementioned paper and build the mathematical expression of displacement and dynamic stress distribution. The solution of the problem is obtained by calculating Green’s function that links the stress distribution to the normal displacements at the interface and, meanwhile, is suitable for various boundary conditions. The advantage of methodology does not need to not only decompose the solution of the problem into quasi-static solution and free vibration solution, but also avoid the complicated integral transformation. It is, therefore, relatively convenient to be conducted by using numerical calculation for analysis of dynamic stress distribution. A case study in [18] is applied to verify the availability and accuracy of the proposed new method. Results illustrate that dynamic stress distributions obtained in this case study are in good agreement with actual data.


## 2. Formulation of Green’s Function

For a thick-walled spherical shell, there is a distinct difference between internal radius and external radius. Internal radius and external radius are set toa and b, respectively, and are under normal impact loads that are ψ
1
(
t
) and ψ
2
(
t
), respectively. Due to the spherical symmetry of the problem, the motion equation of the problem in spherical coordinates can be expressed only by radial displacement component u
(
r
,
t
) and is given by (1)
∂
2
u
∂
t
2
-
c
2
∂
∂
r
1
r
2
∂
∂
r
r
2
u
=
0
a
≤
r
≤
b
,
t
≥
0
,where u is displacement, E is elastic modulus, μ is Poisson’s ratio, r is radius of spherical shell, t is time, and the longitudinal wave velocity is(2)
c
=
E
1
-
μ
ρ
1
+
μ
1
-
2
μ
.Letψ be the normal impact stress; stresses boundary condition of internal radius and external radius at the thick-walled spherical shell are given by(3)
σ
r
r
a
,
t
=
E
1
+
μ
1
-
2
μ
1
-
μ
∂
u
∂
r
+
2
μ
u
r
r
=
a
=
ψ
1
t
,
σ
r
r
b
,
t
=
E
1
+
μ
1
-
2
μ
1
-
μ
∂
u
∂
r
+
2
μ
u
r
r
=
b
=
ψ
2
t
.Initial condition of the problem is(4)
u
r
,
0
=
φ
1
r
,
∂
u
r
,
0
∂
t
=
φ
2
r
.According to Hooke’s law, the relationship between dynamic stress and strain based on polar coordinates can be expressed as(5)
σ
r
r
=
E
1
+
μ
1
-
2
μ
1
-
μ
∂
u
∂
r
+
2
μ
u
r
,
σ
θ
θ
=
σ
φ
φ
=
E
1
+
μ
1
-
2
μ
μ
∂
u
∂
r
+
u
r
,where σ
r
θ
=
σ
r
φ
=
σ
φ
θ
=
0.According to (1) and (3), Green’s function on problem of determining solution can be defined as the following solution of determining solution problem:(6)
∂
2
∂
t
2
-
c
2
∂
∂
r
1
r
2
∂
∂
r
r
2
G
r
,
t
;
ξ
,
τ
=
δ
r
-
ξ
δ
t
-
τ
,
a
≤
r
,
ξ
≤
b
,
t
≥
0
,
1
-
μ
∂
∂
r
+
2
μ
r
G
r
,
t
;
ξ
,
τ
r
=
a
=
0
,
1
-
μ
∂
∂
r
+
2
μ
r
G
r
,
t
;
ξ
,
τ
r
=
b
=
0
,
G
r
,
t
;
ξ
,
τ
t
<
τ
=
0
,
∂
G
r
,
t
;
ξ
,
τ
∂
t
t
<
τ
=
0
.In order to solve above-mentioned equations, it is necessary to introduce two lemmas, which are obtained from impulse principle and in [19], as follows.Lemma 1 (basic integral formula).
Green’s function,G
(
r
,
t
;
ξ
,
τ
), is used to express the solution of determining solution problem about (1) and (3) as follows:(7)
u
r
,
t
=
∫
a
b
φ
2
r
G
r
,
t
;
ξ
,
τ
τ
=
0
-
φ
1
r
∂
G
r
,
t
;
ξ
,
τ
∂
τ
τ
=
0
d
ξ
+
1
ρ
∫
0
t
ψ
2
τ
G
r
,
t
;
ξ
,
τ
ξ
=
b
-
ψ
1
τ
G
r
,
t
;
ξ
,
τ
ξ
=
a
d
τ
.Lemma 2 (impulse principle).
The solution of determining solution problem about (6) is given by(8)
∂
2
∂
t
2
-
c
2
∂
∂
r
1
r
2
∂
∂
r
r
2
G
r
,
t
;
ξ
,
τ
=
0
,
a
≤
r
,
ξ
≤
b
,
t
≥
0
,
1
-
μ
∂
∂
r
+
2
μ
r
G
r
,
t
;
ξ
,
τ
r
=
a
=
0
,
1
-
μ
∂
∂
r
+
2
μ
r
G
r
,
t
;
ξ
,
τ
r
=
b
=
0
,
G
r
,
t
;
ξ
,
τ
t
=
τ
=
0
,
∂
G
r
,
t
;
ξ
,
τ
∂
t
t
=
τ
=
δ
r
-
ξ
.


## 3. Solution of Green’s Function with a Separation Variable Method

It can be seen from Lemmas1 and 2 so long as Green’s function G
(
r
,
t
;
ξ
,
τ
) is obtained that the solution, u
(
r
,
t
), about determining solution problem of (1) and (3), can be easily found. In the meantime, the determining solution problem of (8) and boundary conditions are all due to being homogeneous; they are straightforward to be found by the separation variable method.LetT
=
t
-
τ; the determining solution problem of (8) can be rewritten by(9)
∂
2
G
∂
T
2
-
c
2
∂
∂
r
1
r
2
∂
∂
r
r
2
G
=
0
,
a
≤
r
,
ξ
≤
b
,
t
≥
0
,
1
-
μ
∂
G
∂
r
+
2
μ
r
G
r
=
a
=
0
,
1
-
μ
∂
G
∂
r
+
2
μ
r
G
r
=
b
=
0
,
G
T
=
0
=
0
,
∂
G
∂
t
T
=
0
=
δ
r
-
ξ
.LetG
(
r
,
t
;
ξ
,
τ
)
=
R
(
r
;
ξ
,
τ
)
X
(
t
;
ξ
,
τ
); we get(10)
d
2
X
d
T
2
+
λ
c
2
X
=
0and eigenvalue problem(11)
d
2
R
d
r
2
+
2
r
d
R
d
r
+
λ
2
-
2
r
2
R
=
0
,
(12)
d
R
d
r
+
h
a
R
r
=
a
=
0
,
d
R
d
r
+
h
b
R
r
=
b
=
0
,where (13)
h
=
2
μ
1
-
μ
.Let(14)
j
a
=
λ
j
1
′
λ
a
+
h
a
j
1
λ
a
,
n
a
=
λ
n
1
′
λ
a
+
h
a
n
1
λ
a
,
j
b
=
λ
j
1
′
λ
b
+
h
b
j
1
λ
b
,
n
b
=
λ
n
1
′
λ
b
+
h
b
n
1
λ
b
.Characteristic equation and eigenfunction are, respectively,(15)
j
a
n
b
-
j
b
n
a
=
0
,
(16)
R
r
;
ξ
,
τ
=
n
a
j
1
λ
i
r
-
j
a
n
1
λ
i
r
.Let(17)
J
a
=
h
+
1
J
1
/
2
λ
a
+
h
-
2
J
5
/
2
λ
a
,
Y
a
=
h
+
1
Y
1
/
2
λ
a
+
h
-
2
Y
5
/
2
λ
a
,
J
b
=
h
+
1
J
1
/
2
λ
b
+
h
-
2
J
5
/
2
λ
b
,
Y
b
=
h
+
1
Y
1
/
2
λ
b
+
h
-
2
Y
5
/
2
λ
b
.We get(18)
j
a
=
π
2
λ
a
λ
3
J
a
,
j
b
=
π
2
λ
b
λ
3
J
b
,
n
a
=
π
2
λ
a
λ
3
Y
a
,
n
b
=
π
2
λ
b
λ
3
Y
b
.Based on the relationship between spherical Bessel function and semiodd order function, the characteristic equation (15) can be rewritten by(19)
J
a
Y
b
-
J
b
Y
a
=
0
.Furthermore, let(20)
M
r
=
π
2
λ
r
Y
a
J
3
/
2
λ
r
-
J
a
Y
3
/
2
λ
r
.The eigenfunction (12) is(21)
R
=
λ
3
π
2
λ
a
M
rand its 2-norm is (22)
R
2
=
∫
a
b
r
2
R
2
r
;
ξ
,
τ
d
r
=
∫
a
b
r
2
n
a
j
1
λ
r
-
j
a
n
1
λ
r
d
r
,
R
2
=
N
8
λ
2
a
,where(23)
N
=
2
h
-
1
2
+
4
λ
2
b
2
-
9
λ
2
b
2
J
a
J
b
2
-
2
h
-
1
2
+
4
λ
2
a
2
-
9
λ
2
a
2
.According to principle of linear superposition and orthogonality of Bessel function, we get(24)
G
r
,
t
,
ξ
,
τ
=
∑
i
=
1
∞
C
R
sin
⁡
λ
i
c
t
-
τ
,where(25)
C
=
∫
a
b
r
2
R
i
r
;
ξ
,
τ
δ
r
-
ξ
d
r
λ
i
c
R
1
,
i
2
.LetM
a
=
M
r
r
=
a
,
M
b
=
M
r
r
=
b, and let the thick-walled shell be in a static state when t
=
0; we get(26)
u
r
,
0
=
φ
1
r
=
0
,
∂
u
r
,
0
∂
t
=
φ
2
r
=
0
.In the light of Lemma1, displacement distribution of the thick-walled shell (7) can be expressed by(27)
u
r
,
t
=
4
π
9
ρ
c
∑
i
=
1
∞
M
r
N
∫
0
t
λ
i
2
b
2
ψ
2
t
M
b
-
λ
i
2
a
2
ψ
1
t
M
a
sin
⁡
λ
i
c
t
-
τ
d
τ
.


## 4. Numerical Example

Case 1 (isotropic impact load on spherical external surface).
Let dimensionless variables beT
=
(
c
/
a
)
t, R
=
(
r
-
a
)
/
(
b
-
a
), and σ
i
¯
=
σ
i
/
σ
0, assuming s
=
b
/
a, which is diameter ratio of the thick-walled shell. Function of isotropic impact load with time is (28)
ψ
2
t
=
-
σ
0
H
t
=
0
t
<
0
-
σ
0
t
≥
0
+
,
ψ
1
t
=
0
.
Substituting (20) and (21) into (7), the analytic solution of displacement distribution is given by(29)
u
r
,
t
=
4
π
σ
0
9
ρ
c
2
∑
i
=
1
∞
M
r
λ
i
N
λ
i
2
b
2
M
b
cos
⁡
λ
i
c
t
-
1
.
According to Bessel function, dynamic stress expression can be changed to(30)
σ
r
r
=
4
π
σ
0
27
1
-
μ
∑
i
=
1
∞
λ
i
2
b
2
N
π
2
λ
i
r
M
b
1
+
μ
Y
a
J
1
/
2
λ
i
r
-
J
a
Y
1
/
2
λ
i
r
-
2
-
4
μ
Y
a
J
5
/
2
λ
i
r
-
J
a
Y
5
/
2
λ
i
r
cos
⁡
λ
i
c
t
-
1
,
σ
θ
θ
=
4
π
σ
0
27
1
-
μ
∑
i
=
1
∞
λ
i
2
b
2
N
π
2
λ
i
r
M
b
1
+
μ
Y
a
J
1
/
2
λ
i
r
-
J
a
Y
1
/
2
λ
i
r
+
1
-
2
μ
Y
a
J
5
/
2
λ
i
r
-
J
a
Y
5
/
2
λ
i
r
cos
⁡
λ
i
c
t
-
1
.
Assume material properties of isotropic spherical surfaces and shells are obtained:μ
=
0.25, E
=
200 GPa, and ρ
=
7.8
×
1
0
3 kg/m3. Dynamic stress distribution can be obtained by using the proposed Greens function method.
Figure1 shows radial stress variations with time when the external wall of spherical shell is impacted by instantaneous impact at s
=
2. As shown in Figure 1, there are four various dynamic stress states in three different spread stages of stress wave head. First, the dynamic stress value is zero, when the stress wave head is propagating from one impact wall to the other side, but it did not reach a point of the other side. Second, the dynamic stress value appears as jump discontinuity when the stress wave head has reached points of the other side from one impact wall. Third, the dynamic stress value gently tends to the static stress value under the static load when not only has the stress wave head reached points of the other side from one impact wall, but also it has been far away points of the other side. Last, due to reflection of inner wall, the stress wave presents tensile phenomenon and then rapidly tends to the static stress value under the static load; once the next stress comes to the wall, the dynamic stress value appears as strong jump discontinuity again.Figure 1
Impact radial stress variations of external wall with time (s
=
2).Figure2 shows the radial stress variations along radial points when the external wall of spherical shell is impacted by instantaneous impact at s
=
2. It can be seen from the figure that the radial dynamic stress values present significant change in the internal and external wall points, and the maximum dynamic stress near the inner wall.Figure 2
Impact radial stress variations of external wall with radius (s
=
2).Figures3 and 4 show the radial stress variations with time and the radial stress variations along radial points, respectively, when the external wall of spherical shell is impacted by instantaneous impact at s
=
4. As shown in Figures 3 and 4, with the increase of the cylinder thickness, the radial dynamic stress is first slow tension; second it presents jump with reach of stress wave head; third static stress value will have time extension while the time of reflection wave of the inner wall is lengthened; at last, the maximum compressive stress occurs near the impact wall, while the maximum tensile stress is near the other side.Figure 3
Impact radial stress variations of external wall with time (s
=
4).Figure 4
Impact radial stress variations of external wall with radius (s
=
4).Results from Figures1, 2, 3, and 4 verify that the solutions obtained by using the proposed method are accurate in [12, 18, 20].Case 2 (isotropic impact load on spherical internal surface).
Likewise, function of sudden stress concerning spherical internal surface with time is expressed as follows:(31)
ψ
1
t
=
-
σ
0
H
t
=
0
t
<
0
-
σ
0
t
≥
0
+
,
ψ
2
t
=
0
.
The solution of displacement distribution is given by(32)
u
r
,
t
=
4
π
σ
0
9
ρ
c
2
∑
i
=
1
∞
M
r
λ
i
N
λ
i
2
a
2
M
a
1
-
cos
⁡
λ
i
c
t
.
Using property of Bessel function, dynamic stress expression can be changed to(33)
σ
r
r
=
4
π
σ
0
27
1
-
μ
∑
i
=
1
∞
λ
i
2
a
2
N
π
2
λ
i
r
M
a
1
+
μ
Y
a
J
1
/
2
λ
i
r
-
J
a
Y
1
/
2
λ
i
r
-
2
-
4
μ
Y
a
J
5
/
2
λ
i
r
-
J
a
Y
5
/
2
λ
i
r
1
-
cos
⁡
λ
i
c
t
,
σ
θ
θ
=
4
π
σ
0
27
1
-
μ
∑
i
=
1
∞
λ
i
2
a
2
N
π
2
λ
i
r
M
a
1
+
μ
Y
a
J
1
/
2
λ
i
r
-
J
a
Y
1
/
2
λ
i
r
+
1
-
2
μ
Y
a
J
5
/
2
λ
i
r
-
J
a
Y
5
/
2
λ
i
r
1
-
cos
⁡
λ
i
c
t
.
By using material properties of Case1, dynamic stress distribution can be obtained. For s
=
2, when the internal wall of spherical shell is impacted by instantaneous impact, the radial stress variations with time are shown in Figure 5, and the radial stress variations along radial points are shown in Figure 6. For s
=
4, when the internal wall of spherical shell is impacted by instantaneous impact, the radial stress variations with time are shown in Figure 7, and the radial stress variations along radial points are shown in Figure 8.Figure 5
Impact radial stress variations of external wall with time (s
=
2).Figure 6
Impact radial stress variations of external wall with radius (s
=
2).Figure 7
Impact radial stress variations of external wall with time (s
=
4).Figure 8
Impact radial stress variations of external wall with radius (s
=
4).As shown in Figures5, 6, 7, and 8, when the internal wall of spherical shell is impacted by instantaneous impact, results of radial stress distribution with time and radius have similar results that are obtained when the external wall of spherical shell is impacted by instantaneous impact. The detailed analysis can be found in Case 1. The difference is that when the cylinder is thin, the maximum compression stress is always near the inner wall.


## 5. Discussions and Conclusions

A Green’s function method was conducted in this research to discuss the dynamic stress distribution of thick-walled spherical shell under an isotropic impact load on spherical internal and external surface. Results obtained by the proposed Green’s function method show that radial dynamic stress values have a remarkable change near both internal and external walls. For an isotropic impact load on spherical internal shell, the maximum compressive stress occurs near the impact wall, while the maximum tensile stress is near the other side. For an isotropic impact load on spherical external shell, the maximum compression stress is always near the inner wall. Results also verify that the proposed Green’s function method can be conveniently used to analyze the dynamic stress distribution of thick-walled spherical shell under an isotropic impact load.



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*Source: 169468-2015-12-03.xml*