Considering strain rate effects and deformation magnitude, the gasket material adopts a kinematic hardening combined elastic-plastic constitutive model in the calculation. As the present computational analysis covers a relatively broad range of material parameters, this section provides the parameter ranges of the material model used in the calculations, as listed in Table 1. The specific elastic modulus E and Poisson’s ratio μ adopted in the actual computations will be specified in subsequent sections. Additionally, the influence of gasket material density (ρ) on calculation results is negligible, as demonstrated by theoretical formulas and simulation results in Section 2.3. Therefore, except for the three typical materials specified in Section 2.3 for model validation, all other calculations use a fixed density of 1.5 g/cm³.

In the present computational analysis, the shell material is specified as steel. Based on the same considerations, the constitutive model employed for the shell material in the calculations is tabulated in Table 2.

Considering the requirements for facilitate experimental validation in Section 4.2 and the operational status of actual projectile-borne vulnerable components, this study configures the vulnerable components with typical projectile-borne energetic composite material. This material is characterized by a generalized Maxwell viscoelastic constitutive model, as shown in Figure 3.

The generalized Maxwell visco-elastic model consists of K visco-elastic elements. The relationship between deviatoric strain and deviatoric stress of visco-elastic elements as Equation 1. S i j t = ∫ 0 t 2 G * t − ζ e ˙ i j v e d ζ (1) G * t = G ∞ + ∑ k = 1 N G k ⁡ exp ⁡ − t τ k (2)

In Equation 2, G ^(k) and τ ^(k) represent the shear modulus and relaxation time of the Kth Maxwell body, respectively; G ∞ is the shear modulus at infinite relaxation time.

In the finite element computations, analogous formulation mechanical parameters are employed as substitutes, as tabulated in Table 3. This generalized Maxwell viscoelastic constitutive model for the vulnerable composite materials is implemented in the ANSYS/LS-DYNA explicit dynamics solver for computational analysis via secondary development methodology.

According to classical theory, under axial high-g loading the axial stress in the bottom layer of the protected vulnerable component is (Huang, 2014): σ z = 4 F z π D n 0 2 − D n 2 = 4 a m e π D n 0 2 − D n 2 = D p 2 P m m e M p D n 0 2 − D n 2 (3)

Where, “F _z” is the axial force on the bottom surface of the vulnerable component. “a” represents the acceleration of the entire projectile system. “m _e” denotes the mass of the vulnerable component. “M _p” indicates the projectile mass. “P _m” refers to the single-shot maximum chamber pressure. “D _p” represents the projectile diameter. “D _n” is the outermost diameter of the vulnerable component (in this paper, taken as the vulnerable component diameter D). “D _n0” indicates the innermost diameter of the vulnerable component (in this paper, taken as 0).

Since the calculation result of Equation 3 represents the average axial stress on the bottom surface of the vulnerable component, this article collects data from 20 uniformly distributed stress monitoring points on the bottom surface of the vulnerable component under three typical simulation conditions. It calculates the average axial stress and compares it with the theoretical calculation results of Equation 3, as shown in Table 5. The calculation results indicate that the maximum percentage error between the finite element simulation results and the theoretical prediction results in this article are less than 0.8%. This proves that the simulation model is accurate and effective.

Obtain axial stress data from 20 observation points on the bottom surface of the vulnerable component under three typical simulation conditions, as shown in Figure 5.

As shown in Figure 5, different bottom gasket materials not only affect the magnitude of the bottom layer stress of the vulnerable component, but also affect the axial stress distribution and temporal variation at various monitoring points on the bottom layer of the vulnerable component. More importantly, comparing the stress oscillations in Figures 5b,c, it is found that the compression and release of the gasket interferes with the vibration of the underlying stress of the vulnerable component under impact. This interference results increases in the peak stress of the bottom layer of the vulnerable component. And this effect occurs for gasket materials with better elasticity when overloaded during launch. Therefore, the protection of the vulnerable composite material component by the gasket under the launch load is not achieved through “buffering,” “attenuation” or similar effects.

The Figures 6a,b show the Von Mises stress and axial stress contour charts at the moment of maximum stress in the bottom layer of the vulnerable component under three different working conditions.

The calculation results indicate that the value and distribution of stress in the bottom layer of the vulnerable component vary among three distinct gasket conditions under the same load. Additionally, it is necessary to distinguish and describe different stresses, such as Von Mises stress and axial stress. Therefore, for the convenience of discussion, all stresses mentioned in this article refer to the corresponding values at the moment of maximum stress occurrence. The parameters are introduced such as(σ)_i and σ _max, and their definitions are provided in Table 6. During the discussion, subscripts are added to parameter (z for axial stress, r for radial stress, and τ for circumferential stress), whereas Von Mises stress does not have subscript.

The relationship between the peak stress values and distribution positions of (σ _z)_i and (σ)_i at each monitoring point under three typical working conditions is shown in Figures 7a,b, respectively.

As shown in Figure 7a, the (σ _z)_i distribution at the center of the vulnerable component’s bottom surface remains relatively consistent across the three working conditions. However, notable differences emerge in (σ)_i at the outer edge, primarily due to local stress concentration effects resulting from the deformation of the vulnerable component under stress. This deformation also significantly impacts (σ)_i on the bottom surface of the vulnerable component. As shown in Figure 7b, (σ)_i in the central area of the bottom surface of the vulnerable component is consistently low across all three working conditions. In contrast, (σ)_i on the outer edge is generally higher. Among them, (σ)_i in working condition 3 is consistently higher than that in working conditions 1 and 2. Although (σ)_i under condition 1 is generally lower than that under condition 2, the peak value on the outer side is higher.

As discussed in Section 3.1, significant changes occur in the regularity of force distribution on the bottom layer of the vulnerable component when E of the gasket ranges from 100 to 2000 MPa. Therefore, this E range is crucial for revealing the underlying mechanism. Meanwhile, the difference in regularity between axial stress σ _z and corresponding Von Mises stress σ in Figures 8, 9 indicates that other factors affect σ, leading to its current state. To better understand the protective mechanism of the gasket on the bottom layer of the vulnerable component, we investigated the variations in the value and distribution of the radial stress σ _r and circumferential stress σ _τ in this layer when E of the gasket ranges from 100 to 2000 MPa.

The working condition with μ = 0.4 is selected as the research object. Figure 10a illustrates the value and distribution of σ _z under various working conditions within this range. The contour charts of σ _z of the vulnerable component under various working conditions within this interval are presented in Figures 10b–g. The symmetrical axis of each contour chart is on the left side of the graph. The interface between the vulnerable component and gasket (i.e., the bottom of the vulnerable component) is positioned in the upper region of the graph. The overall positional relationship is consistent with Figure 10a for comparison purposes.

Firstly, as shown in the axial stress contour charts in Figures 10b–g, within the range of 100–2000 MPa for E of the gasket, the peak area (blue part) of the absolute σ _z of the vulnerable component has undergone a shift. At the same time, according to the information in Figure 7a of Section 2.4.2 and Figure 10a, the absolute value of σ _z at the center is higher than that at the outer edge only when E is in the range of 300–1,000 MPa. And in other ranges, the peak σ _z of the bottom layer is always located at its outer edge.

As described in Section 2.4.2, when the range of E is less than 300 MPa or greater than 1,000 MPa, the stress concentration of vulnerable components is mainly near the outer edge. This is due to the difference in radial deformation between vulnerable components and gaskets under axial force. This results in a sharp change in the local shape of the outer edge of vulnerable components. When the E ranges from 300 to 1,000 MPa, the peak stress in the bottom layer of the vulnerable component is observed at the of its the center of bottom. This phenomenon is more akin to the stress distribution on the cross-section of an ideal cylindrical structure under axial loading. Therefore, there is reason to suspect that, within E range of the gasket (between 300 and 1,000 MPa), the existence of a point of E that results in no axial stress concentration in the bottom layer of the vulnerable component at this value. This point is the deformation equilibrium point E _eq described in this article.

The σ _r state of the vulnerable component supports this hypothesis. Figure 11 shows σ _r state of the vulnerable component when E is in the range of 100–2000 MPa (μ = 0.4), which has similar meaning to that of Figure 10. As shown in Figures 11b–g, when E ≤ 300 MPa, σ _r in the bottom layer is positive. This indicates that when E of the gasket is low, the overall radial deformation of the gasket exceeds that of the vulnerable component. This causes the bottom layer of the vulnerable component to become compressed. Based on the data in Figure 11a, a deformation equilibrium point E _eq for the gasket material is proven in the range of 300 MPa < E < 500 MPa (μ = 0.4), which makes σ _r zero. At this time, no axial stress is concentrated in the bottom layer of the vulnerable component. Thus, a reasonable gasket material can play a role in homogenizing the bottom stress of the vulnerable component.

First, it is defined that the “stress coordination function” in this paper specifically refers to the mechanism of reducing von Mises stress through controlled variations in triaxial stresses.

Based on the above conclusions, this section further discusses the regularity of Von Mises stress σ. Figure 12 shows σ state (μ = 0.4) of the vulnerable component when E of the gasket is in the range of 100–2000 MPa. Figures 12a–g has a similar meaning to Figures 10a–g. As shown in Figure 12a, the σ distribution in the bottom layer of the vulnerable component is essentially uniform in the elastic modulus range of 300–500 MPa for the gasket material. According to Figure 8b in Section 3.1, within the elastic modulus range of 20–300 MPa, the overall σ increases, accompanied by local stress concentration at the outer edge. In the elastic modulus range of 300–3 × 10⁵ MPa, the σ gradually increases from the center towards the outer edge.

We discussed the homogenization issue of Von Mises stress in the bottom layer of vulnerable component within the 300∼500 MPa range, and verified that the σ _z distribution of the gasket material at E _eq is uniform (σ _r = 0). Meanwhile, existing theories have shown that an ideal cylinder component experiences circumferential stress σ _r = σ _τ (Huang, 2014) under axial overload conditions. This is consistent with the simulation results presented in this paper, which will not be further elaborated. Therefore, combining the σ formula shows that when the gasket material is near E _eq, its σ value and distribution are primarily determined by the σ _z value and distribution. This causes the similar stress distribution uniformity characteristics in the same interval. The Von Mises stress formula is shown in Equation 4. σ = 2 / 2 σ z − σ r 2 + σ r − σ τ 2 + σ τ − σ z 2 (4)

Similarly, when σ not within the 300–500 MPa range, the influence of the distribution of σ _r and σ _τ must be considered. Typically, under ideal conditions σ _r = σ _τ. Therefore, the variation of vulnerable component deformation, gasket deformation, σ _z and σ _r with E of the gasket were plotted as shown in Figure 13. Among them, the stress coordination point (E _sc) denotes E of the gasket corresponding to the point of minimum σ. Although the minimum value of the simulation result in Figure 8b of Section 3.1 corresponds to E = 1,000 MPa, the selection of simulation calculation parameters is spaced similarly to E _eq. This means that the E _sc can only be determined to have a value ranging from 500 to 2000 MPa (μ = 0.4).

As shown in Figure 13, under the condition of gasket μ = 0.4, the pattern of how σ varies with E of gasket is divided into the following five states.

State (1): When E of the gasket is less than or equal to 100 MPa, the deformation of the gasket increases significantly as its E decreases. This results in a concurrent increase in σ _z at the outer edge of the bottom layer of the vulnerable component. Simultaneously, this deformation causes a concurrent increase in σ _r. It should be noted that in this state, σ _z is negative while σ _r = σ _τ is positive. Therefore, according to the Von Mises stress formula, σ at the bottom layer of the vulnerable component increases as E decreases.

State (2): When 100 MPa < E < E _eq, as E approaches E _eq, the connection between the gasket and the vulnerable component gradually becomes smoother, resulting to a more uniform distribution of σ _z. Simultaneously, σ continues to decrease as E increases, causing the peak position to shift towards the center of the bottom layer of the vulnerable component. At this state, the variation patterns of σ _r and σ _τ are consistent with state (Mohammad and Mohammadzadeh Gonabadi, 2019), and they gradually approach zero as E of the gasket approaches E _eq. Therefore, according to the Von Mises stress formula, σ further decreases as E increases.

State (3): When E = E _eq, the smooth connection between the gasket and the vulnerable component eliminates stress connection, resulting the distribution of σ _z to approximate the ideal stress characteristics of a cylindrical pressure vessel with σ _r = 0 and σ _τ = 0. At this state, the value and distribution of σ are equal to the absolute value of σ _z.

State (4): When E _eq < E ≤ E _sc, E of the gasket gradually moves away from E _eq. The results in the peak position of σ _z shifting outward from the center to the outer edge of the bottom layer of the vulnerable component. Concurrently, peak stress values transition from initial stability to a gradual increase. However, i the radial deformation of the vulnerable component surpasses that of the gasket at this stage. Thus, both σ _r and σ _τ turn negative. As E of the gasket increases, the value of σ _r and σ _τ gradually rise. According to Equation 2, σ actually decreases as E increases, gradually reaching its minimum value. This occurs because the presence of negative values σ _r and σ _τ in Equation 2 diminish the dominant influence of the negative value of σ _z on the outcome of formula.

State (5): When E _sc < E, the deformation capacity of the gasket diminishes as E continues to increase. Concurrently, the deformation of the vulnerable component in relative to the gasket increases and gradually stabilizes at a fixed value. This value corresponds to the deformation of the vulnerable component when the gasket acts as a rigid body. Therefore, as E of the gasket increases, the value of σ _z at the outer edge of the bottom layer also increase, and gradually approaching a fixed value. At the same time, the values of σ _r and σ _τ also increase synchronously and gradually tend to a fixed value. However, in the distribution of σ _r and σ _τ, the values at the outer edge are the smallest, and their subsequent growth rate is significantly lower than that of σ _z. Therefore, according to the Von Mises stress formula, σ continues to rise as E, gradually approaching a fixed value.

Theoretically derivation of the calculation formulas for E _eq and E _sc is challenging due to the adhesive bonding between the bottom layer of the vulnerable component and the gasket. However, the correlation between them and μ of the gasket can be discussed through classical material mechanics. Assuming that the vulnerable component and the gasket are both isotropic elastic cylinders, σ _z on both sides of the interface between the gasket and the vulnerable component is the same, σ _r is equal in magnitude and opposite in direction. According to the Poisson effect, the relationship between the radial strain ε _r and axial strain ε _z of the gasket on the bonding interface between the gasket and the bottom layer of the vulnerable component is as Equation 5. ε r = − μ ε z (5)

Since the σ _z is the same on both sides of the bonding surface between the gasket and the vulnerable component. Therefore, axial strain can be expressed using Equation 6. ε z = σ z / E (6)

Assuming further that the σ _r and radial displacement u _r on the constraint surface are both related to the geometric dimensions of the vulnerable component. We introduce a shear stress coefficient κ, which is assumed to depend on the material properties and geometric shape of the vulnerable component. The σ _r on the constraint surface can be approximately represented follows. σ r ≈ κ · u r = κ · μ σ z D 2 E = κ · 2 μ m a π D E (7)

As mentioned earlier in Section 3.2.2, when E of the gasket material is set to E _eq, the σ _r is zero. Therefore, we assume that |σ _r| is a non-zero minimal value. Based on the left and right limits of the Formula 7, it is not difficult to find that the value of E _eq is related to the properties of the vulnerable component material, and geometric shape, and is directly proportional to Poisson’s ratio μ of the gasket. E _sc is also directly proportional to μ. Figure 14 shows the relationship between material parameters and stress concentration factor. The relationship between E _eq and μ cannot be clearly determine due to the oscillation of axial stress in the bottom layer of the vulnerable component’s material near E _eq, as shown in Figure 14a. However, Figure 14b clearly indicates that as μ of the gasket material increases, so does E of the gasket, This corresponds to the minimum Von Mises stress concentration factor in the bottom layer of the vulnerable component. This suggests that E _sc has shifted in the positive direction of the horizontal axis at this time.

Based on the above discussion, it is established that for given vulnerable component conditions, a specific E _eq necessarily exists for the gasket material’s elastic modulus. When E is set to this value, σ _z and σ at the bottom layer of the vulnerable component are uniformly distributed. At the same time, there exists a E _sc that is slightly greater than E _eq. When E is set to this value, the peak of σ achieves its minimum under the combined effects of various stresses. Crucially, E _eq and E _sc are positively correlated with μ of the gasket material.

The thickness variation of the gasket leads to changes in its longitudinal deformation, impacting the homogenization coordination effect. Therefore, further discussion is needed on the relationship between the thickness H _p of the gasket and the homogenization coordination effect. Take typical gasket thicknesses of H _p = 2, 10, 20 mm, and a material Poisson’s ratio of μ = 0.4. The calculation results under different elastic modulus E are shown in Figure 15.

As shown in Figure 15, when the elastic modulus of the gasket material is small, the thickness of the gasket will significantly increase the longitudinal deformation difference at the interface between the gasket and the grain. The higher the thickness, the greater the stress concentration factor and stress peak. But when the elastic modulus of the gasket is higher than 300 MPa, the regularity of the three curves is consistent. Meanwhile, the optimal elastic modulus value corresponding to the lowest Von Mises stress suggests that the uniform coordination effect of the gasket on the bottom layer stress of the vulnerable component is independent of the thickness of the gasket.