Bending and buckling behavior analysis of foamed metal circular plate

Basic formula

According to the first shear deformation plate theory, the displacement function of any point in the plate is described as follows:

U r ( r ,   z ) = u ( r ) + z ϕ ( r )          Eq. [1]

U z ( r ,   z ) = w ( r )           Eq. [2]

where u and w represent the radial and lateral displacement of the point, respectively. φ represents the rotational angle of middle surface. Equilibrium equations can be deduced by geometrical equations, constitutive equations and the variational principle.

Equilibrium equations and boundary conditions

Equilibrium equations:

{ N θ r − d ( r N r ) r d r = 0 Q r + M θ r − d ( r M r ) r d r = 0 q + d ( r Q r ) r d r − p r ( d 2 w d r 2 + d w d r ) = 0           Eq. [3]

Boundary conditions: N_r = 0, M_r = 0

where N_r, N_θ are radial internal forces; M_r and M_θ are the bending moments; Q_r is the shear force.

Analytical solution of bending deflection for axisymmetric bending

Taking into account the bending of a circular plate with graded foam density and ignoring the radial pressure p, only the uniformly distributed load q is considered and the equilibrium equation of bending can be obtained from [3]:

{ N θ r − d ( r N r ) r d r = 0 Q r + M θ r − d ( r M r ) r d r = 0 q + d ( r Q r ) r d r = 0           Eq. [4]

1. Degraded to the classical results

As long as the lateral shear stiffness is infinite, that means there is no transverse shear deformation. Equation [5] becomes the following results in classic theory:

w = q 64 Ω r 4 − 1 2 + C 3 r 2 + C 4 ( − r 2 2 ln ⁡   r + r 2 4 ) − C 5   ln ⁡   r + C 6           Eq. [5]

C_i (i = 1, 2, …) is integration constant.

2. Degraded to the homogeneous material plate (n = 0, Ω = D)

Taking into account the surrounding clamped circular plate, its boundary conditions are as follows:

when r = b, w = w′ = 0

r = 0, w′ = 0, w & w′ are limited, it can be obtained that

{ C 4 = C 5 = 0 C 3 = q 16 Ω b 2 C 6 = b 2 2 C 3 − q b 4 64 Ω = q b 4 32 Ω − q b 4 64 Ω = q b 4 64 Ω           Eq. [6]

and

w ( r ) = q 64 D ( r 2 − b 2 ) 2           Eq. [7]

The result is exactly the same with the reference (8).

Graded foamed circular plate of bending deflection

Properties of graded foamed metal employ the model appointed by Tang HP, Zhang ZD (9).

E f E s = α ( ρ f ρ s ) m           Eq. [8]

In above formula, E_f is the elastic modulus (usually a function of each point location in the material); E_s is the elastic modulus of the hole wall material (usually a function of each point location in the material); ρ_f is the density of graded foamed metal; ρ_s is the density of the hole wall structure (the density of corresponding solid material); α is a geometric ratio constant.

In order to simplify the calculation process, set α = 1, m = 2, and assuming Poisson’s ratio remains the same, that is

E f E s = ( ρ f ρ s ) 2 v = 0.3           Eq. [9]

It is a symmetric plate with the same relative density on both upper and lower surfaces, called mode I; the density distribution pattern is

ρ f ρ s = n s 2 − n 12 + 1 2           Eq. [10]

It is an asymmetric plate with different relative density of upper electrode, called mode II; the density distribution patter is

ρ f ρ s = n s 2 + ( 1 − n 3 ) s − n 12 + 1 2           Eq. [11]

It can be seen from Figure 2 that the gradient index n has an obvious effect on circular plate bending deformation, and the maximum deflection decreases with the increase of n; with the decrease of the thickness ratio h/b, deflection increases. That is because the plate becomes thicker, the effective stiffness becomes greater; therefore, it is not easy to deform.

It can be concluded from Figure 3 that with the change n from -3 to 3, the deflection of plate center is gradually reduced. It is obvious that the plate stiffness increases, and the center deflection of the plate has a linear relationship with external load. There is a similar trend in Type 1 and Type 2.

Figure 4 shows the bending of circular plate configuration in the case of model I and model II. It is shown that density gradient n has significant effect on the configuration of plate from the above diagram.

For radial fixed simply supported circular plate edges, the boundary conditions are as follows:

r = b , u = w = 0 , M r = 0 , r = 0 , u = w ' = φ = 0

M r = B 11 ( d u d r + ν u r ) + D 11 ( d ϕ d r + ν ϕ r )

Therefore, the integral constant can be obtained as follows:

{ C 1 = B 11 ( 1 + v ) A 11 D 11 ⋅ q b 2 8 C 3 = q b 2 16 Ω H C 2 = C 4 = C 5 = 0 C 6 = q b 4 64 Ω ( 2 H − 1 ) + q b 2 4 A 44 H = 2 Ω ( 1 + v ) D 11 + 1           Eq. [12]

The analytical solution of displacement field of the circular plate is as follows:

u ( r ) = − B 11 A 11 q b 2 16 Ω r [ 1 − ( r b ) 2 ] ​         Eq. [13]

w ( r ) = q b 4 64 Ω [ ( r b ) 4 − 2 H ( r b ) 2 + 2 H − 1 ] + q b 2 4 A 44 [ 1 − ( r b ) 2 ]           Eq. [14]

to [14] dimensionless,

take  w ¯ = w h ,   x ¯ = r b ,  thus  Q = q b 4 h D s ,   F 1 = D s 64 Ω ,   F 2 = D s 4 A 44 b 2 ,   D s = E s h 3 12 ⋅ ( 1 − v 2 )

thus:

w ¯ ( x ) = F 1 Q ( x 4 − 2 H x 2 + 2 H − 1 ) + F 2 Q ( 1 − x 2 )           Eq. [15]

It can be seen from Figure 5 that gradient index n has an obvious effect on the bending deformation of the circular plate, and the maximum deflection decreases with the increase of gradient index n; with the decrease of the thickness ratio h/b, deflection increases. That is because the plate becomes thicker, and the effective stiffness becomes greater. Therefore, it is not easy to deform.

It can be seen in Figure 6, along with the increase of the density gradient, the rigidity of the materials is also increased, but the maximum deflection is reduced (10). The trend of the Type 1 and Type 2 is similar. However, in different boundary conditions, the maximum deflection of simply supported plate is much greater than a fixed one.

It can be seen from Figure 7 that the density gradient n has a significant effect on the configuration of plate. Under different boundary conditions, there is also a certain difference in deflection.