基于ODLV定向量纲的弹性层合板冲击相似律研究

doi:10.7427/S1000-6893.2025.32341

Abstract

Abstract:

Using scaled model experiments to predict the response of full-scale engineering structures under impact loads has the advantages of being economical， efficient， and feasible. As lightweight， high strength composite materials are widely used in large-scale aerospace structures， the study of the impact similitude theory and scaling methods for composite materials is becoming increasingly important. For the similarity scaling problem of elastic laminate under impact response， conventional scaling methods usually fail to work effectively due to the strict requirements imposed by geometric scaling factors and laminate ply orientation matching. This article combines the ODLV directional dimension impact similarity law system with elastic laminate constitutive equation， derives the impact similarity law of elastic laminates， and obtains laminate stiffness coefficient scaling factors which are described by the directional dimension. Based on the similitude relationship， a similitude distortion modified method is proposed. This method adjusts the ply angles of scaled models to achieve stiffness compatibility and modifies the impact velocity to achieve similarity in the elastic laminated plate impact response. It overcomes the scaling constraints inherent in traditional methods and can effectively address both material distortion and width distortion problems. Simulation results show that the scaled models have good similarity with the prototype structure in displacement， impact force， and impact energy response， which suggests that the designed scaled model under the instruction of the laminate impact similarity scaling method proposed in this article can effectively predict the impact dynamic response of the prototype structure.

Key words: oriented dimension, similarity law, similitude distortion, elastic composite laminate, impact response

CLC Number:

V214.3

Yichen WANG, Fei XU, Xinzhe CHANG, Zhaohui CHENG, Wei FENG. Impact similarity law of elastic composite laminate based on ODLV oriented dimension[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(3): 232341.

Figures/Tables 17

Table 1

Oriented dimensions and scale relations in thin-plate impact problem in ODLV system［24-25］

物理量	量纲	相似关系
特征长度 $L ¯ x$	$L x$	$β L x = (L x) m / (L x) p$
特征长度 $L ¯ y$	$L y$	$β L y = (L y) m / (L y) p$
特征长度 $L ¯ z$	$L z$	$β L z = (L z) m / (L z) p$
密度 $ρ$	$M L x - 1 L y - 1 L z - 1$	$β ρ = ρ m / ρ p$
速度 $V z$	$L z T - 1$	$β V z = (V z) m / (V z) p$
正应力 $σ x$	$L x L y - 1 L z - 1 M T - 2$	$β σ x = (β ρ β V z 2) (β L x / β L z) 2$
正应力 $σ y$	$L x - 1 L y L z - 1 M T - 2$	$β σ y = (β ρ β V z 2) (β L y / β L z) 2$
正应力 $σ z$	$L x - 1 L y - 1 L z M T - 2$	$β σ z = (β ρ β V z 2)$
剪应力 $τ x y$	$L z - 1 M T - 2$	$β τ x y = (β ρ β V z 2) / (β L x β L y / β L z 2)$
剪应力 $τ z x$	$L y - 1 M T - 2$	$β τ x z = (β ρ β V z 2) (β L x / β L z)$
剪应力 $τ y z$	$L x - 1 M T - 2$	$β τ y z = (β ρ β V z 2) (β L y / β L z)$
正应变 $ε x$	$L x - 2 L z 2$	$β ε x = (β L z / β L x) 2$
正应变 $ε y$	$L y - 2 L z 2$	$β ε y = (β L z / β L y) 2$
正应变 $ε z$	$1$	$β ε z = 1$
剪应变 $γ x y$	$L x - 1 L y - 1 L z 2$	$β γ x y = (β L z 2 / β L x β L y) 2$
剪应变 $γ x z$	$L x - 1 L z 1$	$β γ x z = (β L z / β L x)$
剪应变 $γ y z$	$L y - 1 L z 1$	$β γ y z = (β L z / β L y)$
时间 $t$	$T$	$β t = (β L z / β V z)$
位移 $δ x$	$L x - 1 L z 2$	$β δ x = β L x (β L z / β L x) 2$
位移 $δ y$	$L y - 1 L z 2$	$β δ y = β L y (β L z / β L y) 2$
位移 $δ z$	$L z$	$β δ z = β L z$
冲击质量 $G$	$M$	$β G = β ρ (β L x β L y β L z)$
法向压力 $P z$	$L x - 1 L y - 1 L z M T - 2$	$β P z = β ρ β V z 2$
法向冲击力 $F z$	$L z M T - 2$	$β F z = β L x β L y β ρ β V z 2$

Table 1

Table 2

Directional dimension of stiffness coefficient for laminate and scale factors

物理量	定向量纲	相似关系
$A 11$	$L x 3 L y - 1 L z - 2 M T - 2$	$β A 11 = β ρ β V z 2 β L x 4 / β L z 3$
$A 22$	$L x - 1 L y 3 L z - 2 M T - 2$	$β A 22 = β ρ β V z 2 β L y 4 / β L z 3$
$A 66$	$L x 1 L y 1 L z - 2 M T - 2$	$β A 66 = β ρ β V z 2 β L x 2 β L y 2 / β L z 3$
$A 12$	$L x 1 L y 1 L z - 2 M T - 2$	$β A 12 = β ρ β V z 2 β L x 2 β L y 2 / β L z 3$
$A 16$	$L x 2 L z - 2 M T - 2$	$β A 16 = β ρ β V z 2 β L x 3 β L y / β L z 3$
$A 26$	$L y 2 L z - 2 M T - 2$	$β A 26 = β ρ β V z 2 β L x β L y 3 / β L z 3$
$B 11$	$L x 3 L y - 1 L z - 1 M T - 2$	$β B 11 = β ρ β V z 2 β L x 4 / β L z 2$
$B 22$	$L x - 1 L y 3 L z - 1 M T - 2$	$β B 22 = β ρ β V z 2 β L y 4 / β L z 2$
$B 66$	$L x 1 L y 1 L z - 1 M T - 2$	$β B 66 = β ρ β V z 2 β L x 2 β L y 2 / β L z 2$
$B 12$	$L x 1 L y 1 L z - 1 M T - 2$	$β B 12 = β ρ β V z 2 β L x 2 β L y 2 / β L z 2$
$B 16$	$L x 2 L z - 1 M T - 2$	$β B 16 = β ρ β V z 2 β L x 3 β L y / β L z 2$
$B 26$	$L y 2 L z - 1 M T - 2$	$β B 26 = β ρ β V z 2 β L x β L y 3 / β L z 2$
$D 11$	$L x 3 L y - 1 M T - 2$	$β D 11 = β ρ β V z 2 β L x 4 / β L z$
$D 22$	$L x - 1 L y 3 M T - 2$	$β D 22 = β ρ β V z 2 β L y 4 / β L z$
$D 66$	$L x 1 L y 1 M T - 2$	$β D 66 = β ρ β V z 2 β L x 2 β L y 2 / β L z$
$D 12$	$L x 1 L y 1 M T - 2$	$β D 12 = β ρ β V z 2 β L x 2 β L y 2 / β L z$
$D 16$	$L x 2 M T - 2$	$β D 16 = β ρ β V z 2 β L x 3 β L y / β L z$
$D 26$	$L y 2 M T - 2$	$β D 26 = β ρ β V z 2 β L x β L y 3 / β L z$

Table 2

Fig.1

Fig.2

Table 3

Mechanical properties of composite materials

材料	$ρ$ /（kg·m^-3）	$E 1$ /GPa	$E 2$ /GPa	$ν 12$	$G 12$ /GPa	$G 13$ /GPa	$G 23$ /GPa
AS4/epoxy^［33］	1 580	126	11	0.28	2.60	2.60	3.90
IM7/977-3^［34］	1 590	162	8.34	0.27	4.96	4.96	4.96
T300/69^［35］	1 600	140	9.00	0.32	4.60	4.60	3.08

Table 3

Table 4

Validation matrix for thickness distortion

编号	$β L y$	材料	铺层层数	单层厚度/mm	厚度畸变度	长宽畸变度
原型	1	AS4	16	0.25	1	1
Md1	1/10	AS4	8	0.05	2	1
Md2	1/10	AS4	6	0.067	2.68	1
Mc3	1/10	AS4	8	0.05	2	1

Table 4

Table 5

Optimization results of ply angle and scale factors for thickness distortion

编号	铺层	$β ρ$	$β V z$	$β t$	$μ f i$	$C V$
原型	［0_2，45₂，-45₂，90₂］_s	1	1	1	1	0
Md1	［0，-45，45，90］_s	1	1.001	0.099 9	1.002	1.0×10^-8
Md2	［13，-49，-72］_s	1	1.001	0.099 9	1.002	1.2×10^-2
Mc3	［0，45，-45，90］_s	1	1	0.1	1	0

Table 5

Table 6

Validation matrix for thickness-material distortion

编号	$β L y$	材料	铺层层数	单层厚度/mm	厚度畸变度	长宽畸变度
原型	1	AS4	16	0.25	1	1
Md4	1/10	IM7	8	0.05	2	1
Md5	1/10	T300/69	8	0.05	2	1
Md6	1/10	IM7	6	0.067	2.68	1
Md7	1/10	T300/69	6	0.067	2.68	1
Mc8	1/10	IM7	16	0.025	1	1

Table 6

Table 7

Optimization results of ply angle and scale factors for thickness-material distortion

编号	铺层	$β ρ$	$β V z$	$β t$	$μ f i$	$C V$
原型	［0₂，45₂，-45₂，90₂］_s	1	1	1	1	0
Md4	［-14，31，-84，-56］_s	1.006	1.120	0.089 3	1.262	3.2×10^-3
Md5	［-14，30，78，-59］_s	1.006	1.050	0.095 2	1.110	1.9×10^-3
Md6	［-35，0，75］_s	1.006	1.137	0.087 9	1.301	7.4×10^-2
Md7	［12，51，71］_s	1.006	1.050	0.095 2	1.110	3.3×10^-2
Mc8	［0₂，45₂，-45₂，90₂］_s	1	1	0.1	1

Table 7

Table 8

Validation matrix for thickness-material-width distortion

编号	$β L y$	材料	铺层层数	单层厚/mm	厚度畸变度	长宽畸变度
原型	1	AS4	16	0.25	1	1
Md9	1/10	AS4	8	0.05	2	1.2
Md10	1/10	AS4	8	0.05	2	1.5
Md11	1/10	IM7	8	0.05	2	1.2
Md12	1/10	IM7	8	0.05	2	1.5

Table 8

Table 9

Optimization results of ply angle and scale factors for thickness-material-width distortion

编号	铺层	$β ρ$	$β V z$	$β t$	$μ f i$	$C V$
原型	［0₂，45₂，-45₂，90₂］_s	1	1	1	1	0
Md9	［-8，-32，-26，-70］_s	1	0.800	0.125 0	0.642	7.7×10^-2
Md10	［-16，16，-9，46］_s	1	0.594	0.111 2	0.355	2.07×10^-2
Md11	［-16，0，90，-57］_s	1.006	0.899	0.111 2	0.813	7.0×10^-2
Md12	［17，-8，0，50］_s	1.006	0.636	0.157 2	0.407	1.1×10^-2

Table 9

Fig.3

Table 10

Thickness distortion scale model： Relative errors of peak displacement， peak force， and length of time

编号	模型	$δ z m a x β δ z / m m$	相对误差/%	$F z m a x β F z / N$	相对误差/%	$T β t / m s$	相对误差/%
原型	AS4	1.09		3 848.4		1.74
Md1	AS4， $η t k = 2$	1.12	4.6	3 763.2	2.2	1.76	1.1
Md2	AS4， $η t k = 2$	1.13	1.8	4 014.7	4.3	1.68	3.4
Mc3	AS4， $η t k = 2$ （传统方法）	1.12	2.8	3 926.5	2.0	1.68	3.4

Table 10

Fig.4

Table 11

Thickness-material distortion scale model： Relative errors of peak displacement， peak force， and length of time

编号	模型	$δ z m a x β δ z / m m$	相对误差/%	$F z m a x β F z / N$	相对误差/%	$T β t / m s$	相对误差/%
原型	AS4	1.09		3 848.4		1.74
Md4	IM7， $η t k = 2$	1.07	1.8	4 096.3	6.4	1.66	4.6
Md5	IM7， $η t k = 2.68$	1.14	4.6	3 869.4	0.5	1.78	2.3
Md6	T300， $η t k = 2$	1.19	9.2	3 845.0	2.0	1.88	8.0
Md7	T300， $η t k = 2.68$	1.08	0.9	3 923.6	<0.1	1.70	2.3
Md8	IM7， $η t k = 1$ （未修正）	0.95	12.8	3 613.9	6.1	1.52	12.6

Table 11

Fig.5

Table 12

Thickness-material-width distortion scale model： Relative errors of peak displacement， peak force， and length of time

编号	模型	$δ z m a x β δ z / m m$	相对误差/%	$F z m a x β F z / N$	相对误差/%	$T β t / m s$	相对误差/%
原型	AS4	1.09		3 848.4		1.74
Md9	AS4， $η w = 1.2$	1.14	4.6	3 763.2	2.2	1.76	1.1
Md10	AS4， $η w = 1.5$	1.07	1.8	4 014.7	4.3	1.68	3.4
Md11	IM7， $η w = 1.2$	1.06	2.8	3 926.5	2.0	1.68	3.4
Md12	IM7， $η w = 1.5$	1.07	1.8	4 129.7	7.3	1.70	2.3

Table 12

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Impact similarity law of elastic composite laminate based on ODLV oriented dimension

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