3.1. Impact stage

It is convenient to introduce a function

(3.7a,b)\begin{equation} q(r,t)=w_t(r,t)+\gamma\varphi(r,0,t)\quad (r< a(t)),\quad q(r,t)=w_t(r,t)\quad (a(t)< r<1),\end{equation}

and present it, in the interval $0< r<1$, by the series

(3.8)\begin{equation} q(r,t)=\sum_{n=1}^\infty q_n(t) \psi_n(r). \end{equation}

Equations (3.5), (3.7a,b) and the orthogonality condition (3.2) provide the coefficients $q_n(t)$,

(3.9)\begin{gather} q_n=\frac{\mbox{d} b_n}{\mbox{d} t}+\gamma\sum_{m=1}^\infty \frac{\mbox{d} b_m}{\mbox{d} t} S_{mn}(a)+\gamma f_n(a), \end{gather}

(3.10a,b)\begin{gather}f_n(a)=\int_0^a \varphi_0(r, 0, a)\psi_n(r)r\,\mbox{d} r,\quad S_{mn}(a)=\int_0^a\varphi_m(r,0,a)\psi_n(r)r\,\mbox{d} r. \end{gather}

The symmetric matrix $\mathbb {S} (a)$ with the elements $S_{nm}(a)$ given by (3.10a,b) is known as the added-mass matrix of a circular elastic plate of radius 1, which is in contact with the liquid only over the region $r< a$. This matrix was derived by Scolan (Reference Scolan2004). The solution of the hydrodynamic problem,

(3.11a–c)\begin{equation} \nabla^2\varphi_n=0\quad (z<0),\quad \varphi_n=0 \quad (z=0, r>a),\quad \varphi_{nz}=\psi_n(r)\quad (z=0, r< a), \end{equation}

where $n\geq 0$ and $\psi _0(r)=-1$, reads

(3.12a,b)\begin{equation} \varphi_n(r,0,a)=a\int_r^a \frac{\chi_n(\xi/a)\,\mbox{d}\xi}{\sqrt{\xi^2-r^2}},\quad \chi_n(\mu)=\frac{2}{\rm \pi}\int_0^{\mu} \frac{\sigma\psi_n(a\sigma)\,\mbox{d}\sigma}{\sqrt{\mu^2-\sigma^2}}. \end{equation}

The solution (3.12a,b) is convenient to present as

(3.13)\begin{equation} \left.\begin{gathered} \chi_n(\mu)=\frac{2}{\rm \pi}\mu Q_n(a\mu),\quad \varphi_n(r,0,a)=\frac{2}{\rm \pi}\int_r^a \frac{\xi Q_n(\xi)\,\mbox{d}\xi}{\sqrt{\xi^2-r^2}},\\ Q_n(\xi)=\int_0^{{\rm \pi}/{2}} \psi_n(\xi\sin\theta)\sin\theta\,\mbox{d} \theta. \end{gathered}\right\} \end{equation}

Then it is easy to show that

(3.14a,b)\begin{equation} Q_0(\xi)={-}1,\quad \varphi_0(r,0,a) ={-}\frac{2}{\rm \pi}\sqrt{a^2-r^2}, \end{equation}

where $0< r< a$, and

(3.15)\begin{equation} S_{nm}(a)=\frac{2}{\rm \pi}\int_0^{a} x^2 Q_n(x)Q_m(x)\,\mbox{d}\kern0.7pt x. \end{equation}

The functions $Q_n(a)$ and $S_{nm}(a)$ are oscillating in the interval $0\le a\le 1$. The calculations revealed that the derivative $Q'_n(a)$ changes its sign $n$ times in this interval. In each subinterval, where $Q'_n(a)$ has a certain sign, the function $Q_n(a)$ can be approximately presented by 25 points. In total, the function $Q_n(a)$ is presented by $25n$ points with distance $1/(25 n)$ between the points. The derivatives of the added-mass elements, $S'_{nm}(a)=(2/ {\rm \pi}) a^2 Q_n(a)Q_m(a)$, and $S_{nm}(a)$ as well, are also oscillating. The elements $S_{nm}(a)$ for $m\le n$ can be approximately presented with $25n$ points, which are the same as for $Q_n(a)$. Between these points, the functions $Q_n(a)$ and $S_{nm}(a)$ are approximated linearly. By using this approximations, the functions $Q_n(a)$ and $S_{nm}(a)$ are precalculated numerically for $a$ from zero to one with step $0.002$ for $1\leq n\leq m\leq 20$. The added-mass matrix is constant $\mathbb {S} (1)$ at the penetration stage, when the elastic cone is completely wetted.

Equation (2.13) for the normal displacement $w(r,t)$ of the shell elements can be written now as

(3.16)\begin{equation} \frac{\partial q}{\partial t}={-}\frac{1}{\alpha} \left\{\nabla^4w+\frac {\delta}{r^2}(w+ \nu r u_r+u)\right\}. \end{equation}

Substituting (3.8), (3.5), (2.17) and (2.18) into (3.16), multiplying both sides of the equation by $r\psi _m(r)$, and integrating the result with respect to $r$ from $r=0$ to $r=1$ with account for (3.1a,b), we obtain

(3.17)\begin{equation} \frac{\mbox{d} q_m}{\mbox{d} t}={-}\frac{1}{\alpha}\left\{\lambda_m^4 b_m + \delta\sum_{n=1}^\infty b_n(t) T_{mn}\right\}, \end{equation}

where

(3.18)\begin{align} T_{mn} &= \frac 12 (1+\nu)^2 \int_0^1 \psi_n(r)\,\mbox{d} r \int_0^1 \psi_m(r)\,\mbox{d} r +(1-\nu^2) \left\{\int_0^1 \psi_m(r)\psi_n(r)\frac{\mbox{d} r}{r} \right.\nonumber\\ &\quad \left.-\frac{1}{2}\int_0^1 \psi_m(r)\int_0^r \psi_n(r_0)\mbox{d} r_0\frac{\mbox{d} r}{r^2}-\frac{1}{2}\int_0^1 \psi_n(r)\int_0^r \psi_m(r_0)\mbox{d} r_0\,\frac{\mbox{d} r}{r^2}\right\} . \end{align}

The matrix $\mathbb {T}$ with the elements $T_{nm}$ given by (3.18) is symmetric and depends only on the Poisson ratio $\nu$ of the shell material. This matrix is precalculated numerically using the orthonormal functions (3.3).

Equations (3.9) and (3.17) and the initial conditions (2.8) can be written in the matrix form,

(3.19)\begin{equation} \left.\begin{gathered} \frac{\mbox{d} \boldsymbol{b}}{\mbox{d} t}=(\mathbb{I} +\gamma \mathbb{S}(a))^{{-}1}(\boldsymbol{q}-\gamma\boldsymbol{f}(a))\quad \frac{\mbox{d} \boldsymbol{q}}{\mbox{d} t}={-}\frac{1}{\alpha}(\mathbb{D}+\delta\mathbb{T})\boldsymbol{b},\\ \boldsymbol{b}(0)=0, \quad \boldsymbol{q}(0)=0, \end{gathered}\right\} \end{equation}

where $\mathbb {D}= \textrm {diag} (\lambda _1^4,\lambda _2^4,\lambda _3^4,\ldots )$ is a diagonal matrix, $\boldsymbol{b}(t)=(b_1,b_2,b_3,\ldots )^\textrm {T}$, $\boldsymbol{q}(t)=(q_1,q_2,q_3,\ldots )^\textrm {T}$ and $\boldsymbol{f}(a)=(f_1,f_2,f_3,\ldots )^\textrm {T}$.

The radius $a(t)$ of the wetted part of the cone during the impact stage is calculated using the Wagner condition that the elevation of the liquid free surface is equal to the vertical coordinate of the deformed surface of the cone at $r=a(t)$. This condition was reduced by Scolan (Reference Scolan2004) to the equation, which in the dimensionless variables (2.10a–d) has the form

(3.20)\begin{equation} a(t)=\frac4{\rm \pi} \left(t-\sum_{n=1}^\infty b_n(t) Q_n(a)\right). \end{equation}

The system (3.19) and (3.20) is integrated in time up to a time instant $t_*$, when $a(t_*)=1$. Note that the dimensionless duration of the impact stage for a rigid cone is equal to ${\rm \pi} /4$. Indeed, for a rigid cone, we should set $b_n=0$ in (3.20), which gives $a(t)=4t/{\rm \pi}$ and $a=1$ at $t={\rm \pi} /4$.

3.2. Penetration stage and wet/dry modes

For $t>t_*$, when the shell is completely wetted but continues to penetrate into the liquid, we set $a=1$, $\boldsymbol{f} =\boldsymbol{f}(1)$, $\mathbb {S}=\mathbb {S}(1)$ and continue numerical integration of the system (2.3) in time using the matching conditions,

(3.21a,b)\begin{equation} \boldsymbol{b}(t_*+0)=\boldsymbol{b}(t_*-0),\quad \boldsymbol{q}(t_*+0)=\boldsymbol{q}(t_*-0), \end{equation}

where $\boldsymbol{b}(t_*-0)$ and $\boldsymbol{q}(t_*-0)$ are the solutions of the initial problem (3.19) at the end of the impact stage.

It is convenient to introduce a new unknown vector $\boldsymbol{g}(t)=\boldsymbol{q}(t)-\gamma \boldsymbol{f}(1)$. Then the system (3.19) at the penetration stage and the matching conditions (3.21a,b) read

(3.22)\begin{equation} \left.\begin{gathered} \frac{\mbox{d} \boldsymbol{b}}{\mbox{d} t}=(\mathbb{I} +\gamma \mathbb{S}(1))^{{-}1}\boldsymbol{g},\quad \frac{\mbox{d} \boldsymbol{g}}{\mbox{d} t}={-}\frac{1}{\alpha}(\mathbb{D}+\delta\mathbb{T})\boldsymbol{b}, \\ \boldsymbol{b}(t_*+0)=\boldsymbol{b}(t_*-0),\quad \boldsymbol{g}(t_*+0)=\boldsymbol{q}(t_*-0)-\gamma\boldsymbol{f}(1). \end{gathered}\right\} \end{equation}

The homogeneous system (3.22) of ordinary differential equations without the matching conditions at $t=t_*$ has a periodic solution $\boldsymbol{b}(t)=\boldsymbol{B} \cos (\varOmega t)$, where the constant vector $\boldsymbol{B}$ is a solution of the system

(3.23)\begin{equation} ((\mathbb{I} +\gamma \mathbb{S}(1))^{{-}1}(\mathbb{D}+\delta\mathbb{T}) -\mu\mathbb{I})\boldsymbol{B}=0. \end{equation}

Non-zero solutions of the system (3.23) with symmetric matrix $(\mathbb {I} +\gamma \mathbb {S}(1))^{-1}(\mathbb {D}+ \delta \mathbb {T})$ exist only if $\mu =\alpha \varOmega ^2$ is an eigenvalue of this matrix. The eigenvalues are numbered in such a way that $\mu ^{(w)}_n<\mu ^{(w)}_{n+1}$ where $n\geq 1$ and the superscript $w$ stands for wet conditions. The eigenvalues depend on parameters $\gamma$ and $\delta$ and are independent of the shell rigidity and speed of the impact. The corresponding solutions $\boldsymbol{B}_n^{(w)}$ of the system (3.23) for $\mu =\mu _n^{(w)}$ provide the so-called wet modes,

(3.24)\begin{equation} W^{(w)}_n(r)=\sum_{j=1}^\infty B_{nj}^{(w)} \psi_j(r),\end{equation}

of the completely wetted conical shell. Note that the determinant of the matrix of system (3.23) at $\mu =\mu _n^{(w)}$ is zero. To find the vector $\boldsymbol{B}_n^{(w)}$, we truncate the system retaining $(N_{mod}-1)$ equation and setting $B_{n,j}^{(w)}=0$ for $j>N_{mod}$ and $B_{n,N_{mod}}^{(w)}=1$. Finally, the vector $\boldsymbol{B}_n^{(w)}$ is rescaled in such a way that $B_{n,n}^{(w)}=1$. The wet modes $W^{(w)}_n(r)$ are orthogonal but not orthonormal,

(3.25)\begin{equation} \int_0^1 W^{(w)}_n(r) W^{(w)}_m(r) r\,{\rm d} r=\eta_n^2\delta_{nm},\end{equation}

where $\eta _n^2=(\boldsymbol{B}_{n}^{(w)}\boldsymbol{B}_{n}^{(w)})$. The wet modes of partially wetted cone are different from the modes of the totally wetted cone and are not considered in the present analysis.

The dimensionless angular frequency $\varOmega _n^{(w)}$ of the wet $n$th mode $W^{(w)}_n(r)$ is equal to $\varOmega _n^{(w)}=(\mu ^{(w)}_n/\alpha )^{1/2}$. The corresponding dimensional angular frequency of the $n$th wet mode is equal to $\omega _n^{(w)}=\varOmega _n^{(w)}/T_{sc}=(\mu ^{(w)}_n/\alpha )^{1/2}/T_{sc}=[\mu _n^{(w)} D/(\rho _s h R^4)]^{1/2}$. The ratio of the duration of the impact stage $T_{(im)}$, which is equal to ${\rm \pi} /4T_{sc}$ for constant impact speed and rigid cone, to the period of the first wet mode, $T_1^{(w)}=2{\rm \pi} /\omega _1^{(w)}$, is equal to

(3.26)\begin{equation} \frac{T_{(im)}}{T_1^{(w)}}={\frac{1}{8}\varOmega_1^{(w)}}= \frac{1}{8} \sqrt{\frac {\mu^{(w)}_1}{\alpha}}. \end{equation}

This ratio is inverse proportional to the impact speed $V$, see the definition of the parameter $\alpha$ in (2.16a–d). The smallest eigenvalue $\mu ^{(w)}_1$ of the matrix in (3.23), which appears in (3.26), depends on the deadrise angle $\beta$, length of the cone $R$ and thickness $h$ of the shell through the parameter $\delta$ and on the density of the shell material $\rho _s$ and the density of the liquid $\rho$ through the parameter $\gamma$ and is independent of the shell rigidity.

The time ratio (3.26) is used to identify the regime of the hydroelastic impact (see Faltinsen (Reference Faltinsen1999, Reference Faltinsen2000) and Malenica et al. (Reference Malenica, Gatin, Seng, Jagite, Diebold, Khabakhpasheva and Korobkin2021, Reference Malenica, Seng, Diebold, Scolan, Korobkin and Khabakhpasheva2022)). If the ratio is large, this is the impact duration is much longer than the period of the lowest wet mode, then the elastic behaviour of en elastic structure during water impact is well described by the so-called quasi-static decoupled approximation. Within this approximation, one calculates the hydrodynamic pressures acting on an equivalent rigid cone and then applies these pressures to the elastic structure without account for the structural inertia, see § 1.1. For small time ratio (3.26), see figure 1, the impact duration $T_{(im)}$ is short and the maximum deflections and stresses occur at the penetration stage, where the lowest wet modes $W_n^{(w)}(r)$ provide main contributions, see Korobkin & Khabakhpasheva (Reference Korobkin and Khabakhpasheva2006).

By using the wet modes $W_n^{(w)}(r)$, the equations (3.22) can be explicitly integrated. During the penetration stage, the deflection of the conical shell can be presented as the series with respect to wet modes:

(3.27)\begin{equation} w(r,t)= \sum_{n=1}^\infty C_n(t)W_n^{(w)}(r), \end{equation}

instead of the series (3.5), which is with respect to the simplified modes. The coefficients $C_n(t)$ are related to the coefficients $b_n(t)$ in (3.5) by

(3.28)\begin{equation} \boldsymbol{b}=\mathbb{B}^{(w)} \boldsymbol{C}, \end{equation}

where $\boldsymbol{C}(t)=(C_1,C_2,C_3,\ldots )^\textrm {T}$ and the columns of the matrix $\mathbb {B}^{(w)}$ are the vectors $\boldsymbol { B_n}^{(w)}$. Then the system (3.22) leads to the following equations for the coefficients $C_n(t)$, $n\le 1$,

(3.29)\begin{equation} \frac{\mbox{d}^2 C_n}{\mbox{d} t^2} +(\varOmega_n\eta_n)^2C_n=0\quad (t>t_*). \end{equation}

The matching conditions (3.21a,b) provide

(3.30)\begin{gather} \boldsymbol{C}_n(t_*+0)=(\mathbb{B}^{(w)})^{\rm T}\boldsymbol{b}(t_*-0), \end{gather}

(3.31)\begin{gather}\frac{\mbox{d} C_n}{\mbox{d} t} (t_*+0)= (\mathbb{B}^{(w)})^{\rm T} (\mathbb{I}+\gamma\mathbb{S}(1))^{{-}1}\boldsymbol{g}(t_*-0). \end{gather}

Therefore, the formulae, $n\ge 1$,

(3.32)\begin{equation} C_n(t)=C_n (t_*+0)\cos[\varOmega_n\eta_n(t-t_*)]+ \frac{\mbox{d} C_n}{\mbox{d} t}(t_*+0)\frac{1}{\varOmega_n\eta_n} \sin [\varOmega_n\eta_n(t-t_*)], \end{equation}

yield explicit solution of the initial problem (3.29)–(3.31) for the penetration stage.

If the elastic cone is not in contact with water, then its natural oscillations are also described by the system (3.23) but now with $\gamma =0$. The corresponding eigenvalues $\mu ^{(d)}_n$ and solutions $\boldsymbol{B}_n^{(d)}$ are denoted with superscript ($d$) and calculated in the same way as for the wet modes. The corresponding dimensional angular frequency of the $n$th dry mode is equal to $\omega _n^{(d)}=\varOmega _n^{(d)}/T_{sc}=(\mu ^{(d)}_n/\alpha )^{1/2}/T_{sc}=[\mu _n^{(d)} D/(\rho _s h R^4)]^{1/2}$, where $\mu _n^{(d)}$ depends only on the parameter $\delta$. Note that the ratios $\omega _n^{(w)}/\omega _n^{(d)}$ are independent of the shell rigidity.

The dry modes of an elastic conical shell are not used in the present algorithm. They are calculated together with the corresponding natural frequencies as part of the numerical solution of the hydroelastic fully nonlinear slamming problems (see Malenica et al. Reference Malenica, Seng, Diebold, Scolan, Korobkin and Khabakhpasheva2022) within the modal approach. In this approach the displacements of an elastic shell are sought as a combination of the dry modes of the shell with unknown time-dependent coefficients. The matrix equation of structural dynamics with respect to these coefficients is much simpler than the original finite-element equation. This matrix equation is solved together with the discretised equations of computational fluid dynamics using an iterative algorithm with an under-relaxation scheme. It is known that iterations do not converge without such a scheme for hydroelastic problems with a strong added-mass effect (see Seng, Jensen & Malenica (Reference Seng, Jensen and Malenica2014) and Causin, Gerbeau & Nobile (Reference Causin, Gerbeau and Nobile2005)). One of the options for numerical solving of hydroelastic slamming problem is to ‘… subtract an added mass force from both sides of the [structural] equation… Then the [structural] acceleration is obtained by dividing the resulting force by the sum of a [structural] mass and a first approximation of the added-mass [matrix]’, see the ‘Fluid–structure coupled analysis’ in Sun & Faltinsen (Reference Sun and Faltinsen2006). We suggest to use the added-mass matrix calculated within the Wagner model of hydroelastic slamming as the ‘first approximation of the added-mass matrix’ mentioned by Sun & Faltinsen (Reference Sun and Faltinsen2006). This added-mass matrix for dry modes of the structure can be obtained using the added-mass (3.15) for simplified structure and the solutions $\boldsymbol{B}_n^{(d)}$, $\mu _n^{(d)}$ of the system (3.23) for $\gamma =0$. Such a combination of numerical and analytical approaches is under investigation at present.

3.3. Hydrodynamic loads

The present approach, where the system (3.19) is integrated in time during the impact stage with $a(t)<1$ and the system (3.22) is integrated in time during the penetration stage with $a(t)=1$, does not require explicit evaluation of the hydrodynamic loads. The loads are difficult to compute and measure in problems of hydrodynamic slamming (see Faltinsen (Reference Faltinsen1999, Reference Faltinsen2000) and Malenica et al. (Reference Malenica, Seng, Diebold, Scolan, Korobkin and Khabakhpasheva2022)). The pressure $p(x,0,t)$ can be reconstructed from the equation for the normal displacements of the shell elements (2.13). In the dimensionless variables (2.10a–d), we have $p(r,0,t)=-\varphi _t(r,0,t)$. Substituting the series for the normal displacements (3.5) into the dimensionless shell equation (2.13), and using (3.7a,b) and (3.16), we find

(3.33)\begin{gather} p(r,0,t)=\sum_{n=1}^\infty p_n(t) \psi_n(r), \end{gather}

(3.34)\begin{gather}p_n(t)=\frac{1}{\gamma}\frac{\mbox{d}^2 b_n}{\mbox{d} t^2}+ \frac{1}{\alpha\gamma}\left\{\lambda_n^4 b_n +\delta\sum_{m=1}^\infty b_m(t) T_{nm}\right\}. \end{gather}

Note that $0< r<1$ in (3.33), and $p(r,0,t)=0$, where $a(t)< r<1$ during the impact stage. The second derivatives $\mbox {d}^2 b_n/\mbox {d} t^2$ in (3.33) are calculated for the impact stage using the system

(3.35)\begin{equation} (\mathbb{I} +\gamma \mathbb{S}(a))\frac{\mbox{d}^2 \boldsymbol{b}}{\mbox{d} t^2}= \frac{\mbox{d} \boldsymbol{q}}{\mbox{d} t} +\gamma\frac{\mbox{d} a}{\mbox{d} t} \left\{\frac{2 a^2}{\rm \pi}\boldsymbol{Q}- \mathbb{S}'(a)\frac{\mbox{d} \boldsymbol{b}}{\mbox{d} t}\right\}, \end{equation}

which is obtained by differentiation (3.19) with respect to time. Here $\mbox {d} a/\mbox {d} t=0$ at the penetration stage. Note that (3.10a,b), (3.13) and (3.14a,b) give

(3.36)\begin{equation} f'_n(a)={-}\frac{2 a^2 }{\rm \pi}Q_n(a). \end{equation}

In (3.35), the derivatives $\mbox {d} \boldsymbol {b}/\mbox {d} t$ and $\mbox {d} \boldsymbol {q}/\mbox {d} t$ are calculated using (3.19), the derivative $\mbox {d} a/\mbox {d} t$ is obtained by differentiation (3.20) with respect to time,

(3.37)\begin{equation} \frac{\mbox{d} a}{\mbox{d} t}=\frac{1-\boldsymbol{Q}(a)\boldsymbol{\cdot} \mbox{d} \boldsymbol{b}/\mbox{d} t}{{\rm \pi}/4+\boldsymbol{Q}'(a)\boldsymbol{\cdot}\boldsymbol{b}}, \end{equation}

where $\boldsymbol{Q}(a)=(Q_1(a), Q_2(a), Q_3(a),\ldots )^\textrm {T}$, $\boldsymbol{Q}'(a)=(Q'_1(a), Q'_2(a), Q'_3(a),\ldots )^\textrm {T}$ and $Q_n(a)$ are given by (3.13), and

(3.38a,b)\begin{equation} Q'_n(\xi)=\int_0^{{\rm \pi}/{2}} \psi'_n(\xi\sin\theta)\sin^2\theta\,\mbox{d} \theta,\quad S'_{nm}(a)=\frac{2}{\rm \pi} a^2 Q_n(a)Q_m(a). \end{equation}

On the other hand, the linearised Bernoulli equation, $p(r,0,t)=-\varphi _t(r,0,t)$, the series (3.5) for the velocity potential, and the formulae (3.13), (3.14a,b) and (3.37) provide

(3.39)\begin{gather} p(r,0,t)=\frac{B(t)}{\sqrt{a^2-r^2}}-\sum_{n=1}^\infty \frac{\mbox{d}^2 b_n}{\mbox{d} t^2}\varphi_n(r,0,a), \end{gather}

(3.40)\begin{gather}B(t)=\frac 2{\rm \pi} a\dot a \left(1-\sum_{n=1}^\infty \frac{\mbox{d} b_n}{\mbox{d} t} Q_n(a)\right). \end{gather}

Equations (3.13) yield that $\varphi _n(r,0,t)=O(\sqrt {a^2-r^2})$ as $r\to a$ and $r< a$. We conclude that the pressure is square-root singular at the contact line $r=a(t)$ during the impact stage, $p(r,0,t)=O(1/\sqrt {a^2-r^2})$, and behaves as $p(r,0,t)=O(\sqrt {1-r^2})$ during the penetration stage, where $r\to 1-0$. As a result, the series (3.33) converges slowly when $a(t)<1$. To improve the convergence, we decompose the series (3.33) during the impact stage as

(3.41)\begin{equation} p(r,0,t)=B(t)\frac{H(a^2-r^2)}{\sqrt{a^2-r^2}}+\sum_{n=1}^\infty \tilde{p}_n(t)\psi_n(r), \end{equation}

where $H(x)$ is the Heaviside step function, $H(x)=1$ for $x\ge 0$ and $H(x)=0$ for $x<0$. Multiplying both sites of (3.41) by $\psi _m(r)r$ and integrating the results with respect to $r$ from zero to one using the orthogonality relation (3.2), we find

(3.42)\begin{equation} \tilde{p}_m(t)=p_m(t)-aB(t)Q_m(a), \end{equation}

during the impact stage where $a(t)<1$. During the penetration stage, we have $a(t)=1$ and $B(t)=0$ in (3.42). Coefficients $\tilde {p}_m$ decay quicker as $m\to \infty$ than $p_m(t)$ given by (3.33).

Note that the pressure in the problem of rigid cone impact reads

(3.43)\begin{equation} {p^{(R)}(r,0,t)}=\frac{8a}{{\rm \pi}^2}\frac{H(a^2-r^2)}{\sqrt{a^2-r^2}}.\end{equation}

The dimensionless hydrodynamic force $F(t)$ acting on the entering elastic cone is obtained using (3.41), as

(3.44)\begin{equation} F(t)=2{\rm \pi}\int_0^{a(t)} p(r,0,t) r\,\mbox{d} r = 2{\rm \pi} a B(t)H(1-a)+2{\rm \pi} \sum_{n=1}^\infty \tilde{p}_n(t) \int_0^{a(t)}\psi_n(r) r\,\mbox{d} r, \end{equation}

where the scale of the force is $\rho V^2 R^2$.

4.1. Wet/dry modes and their frequencies

The angular frequencies of the wet, $\omega ^{(w)}$, and dry, $\omega ^{(d)}$, modes of elastic conical shell are obtained using the roots $\mu _n$, where $\mu _{n+1}>\mu _{n}$ and $n\ge 1$, of the equation,

(4.1)\begin{equation} \det ((\mathbb{I} +\gamma \mathbb{S}(1))^{{-}1}(\mathbb{D}+\delta\mathbb{T}) -\mu\mathbb{I})=0,\end{equation}

with $\gamma >0$ and $\gamma =0$ correspondingly for the conditions of case 2. The dry, $\omega _n^{(d)}/2{\rm \pi}$, and wet $\omega _n^{(w)}/2{\rm \pi}$, frequencies as functions of the cone deadrise angle $\beta$ are shown in figure 4(a) and 4(b). To calculate four lowest frequencies, the matrix $((\mathbb {I} +\gamma \mathbb {S}(1))^{-1}(\mathbb {D}+\delta \mathbb {T}) -\mu \mathbb {I})$ is truncated retaining 20 terms and its determinant is calculated as a function of $\mu$ with a certain step starting from $\mu =0$. The intervals, where the determinant changes its sign, are identified and the bisection method is used within each interval to find the roots $\mu _n$ with a required accuracy of $10^{-10}$.

Figure 4. The frequencies of dry (a) and wet (b) modes with $n=1,2,3,4$ as functions of the cone deadrise angle $\beta$ for case 2. The ratios of wet and dry frequencies for lowest modes with $n=1,2,3,4$ (c) and the deviations of the ratios from their mean values (d) in percentage.

The ratios of the corresponding wet and dry frequencies, see figure 4(c), weakly depend on the cone deadrise angle. The mean values ${\rm \Delta} w_n^{(av)}$ of the ratios in the interval $0\le \beta \le 15^\circ$ are ${\rm \Delta} \omega _1^{(av)}= 0.396$, ${\rm \Delta} \omega _2^{(av)}= 0.55$, ${\rm \Delta} \omega _3^{(av)}= 0.624$ and ${\rm \Delta} \omega _4^{(av)} = 0.683$. The relative deviations of the ratios from their mean values,

(4.2)\begin{equation} {\rm \Delta}\omega_n=\frac{\omega^{(w)}_n(\beta)/\omega^{(d)}_n(\beta)- {\rm \Delta}\omega_n^{(av)}}{{\rm \Delta}\omega_n^{(av)}} \cdot 100\,\%, \end{equation}

are shown in figure 4(d) for $n=1$, 2, 3 and 4. We conclude that both dry and wet frequencies strongly depend on the cone deadrise angle, but their ratios are approximately constant in the interval $0\le \beta \le 15^\circ$. Here $\beta =0$ corresponds to the circular elastic plate, which is clamped at the centre and at the edge. The obtained results about weak dependence of the ratios of dry and wet frequencies on the cone deadrise angle are similar to the findings by Korobkin et al. (Reference Korobkin, Khabakhpasheva and Shishmarev2023) for the ratios of dry and wet frequencies of two-dimensional floating complex elastic structures.

The frequencies of the three lowest dry modes of the cone with deadrise angle $\beta =10^\circ$ were computed with the help of the finite-element structural solver ABAQUS by Malenica et al. (Reference Malenica, Seng, Diebold, Scolan, Korobkin and Khabakhpasheva2022) as $\omega _1^{(d)}/2{\rm \pi} = 247$ Hz, $\omega _2^{(d)}/2{\rm \pi} = 334.6$ Hz and $\omega _3^{(d)}/2{\rm \pi} = 475.9$ Hz. The frequencies $\omega _1^{(d)}/2{\rm \pi} = 247.25$ Hz, $\omega _2^{(d)}/2{\rm \pi} = 336.2$ Hz and $\omega _3^{(d)}/2{\rm \pi} = 479$ Hz predicted by the present model, see figure 4(a), are very closed to the computed ones.

The shapes of the dry and wet modes of the elastic cone for case 2, as well as the shapes of the circular plate modes are shown in figure 5 for $n=1,2,3$ and $n=10$. It is seen that the higher modes are weakly dependent on the deadrise angle and the presence of the liquid in contact with the structure.

Figure 5. Shapes of the elastic modes in case 2 for $\beta =10^\circ$ with $n=1$ (a), $n=2$ (b), $n=3$ (c) and $n=10$ (d). The blue lines are for dry modes $\psi _n(r)$ of equivalent circular plate, black lines are for dry modes $W^{(d)}_n(r)$ and red lines are for wet modes $W^{(w)}_n(r)$ of the conical shell.

The eigenfrequencies of a conical shell strongly depend on the deadrise angle of the cone. Figure 4(a) shows that the lowest dry frequency $\omega _1^{(d)}/2{\rm \pi}$ of the circular plate is 56.5 Hz and the frequencies of the conical shell with $\beta =5^\circ$ and $10^\circ$ are 141.5 and 247.25 Hz correspondingly. We conclude that a conical shell is much less flexible than an equivalent circular plate with the same elastic characteristics even for small deadrise angles. It is important to mentioned that dry and wet lowest modes of the cone, see figure 5, are not close to each other as is the case for two-dimensional problems of hydroelasticity, see Korobkin et al. (Reference Korobkin, Khabakhpasheva and Shishmarev2023).

4.2. Deflections and strains during the conical shell impact on water

Calculations are performed for case 1 to compare the deflections of the elastic cone within the Scolan model and the present model. Note that the results by Scolan (Reference Scolan2004) are provided by the present model with $\delta =0$. The system (3.19) and (3.20) is integrated with respect to the dimensionless radius of the wetted surface $a(t)$ using the Runge–Kutta fourth-order method with step ${\rm \Delta} a=10^{-4}$. The system is integrated with 8 and 20 modes to confirm convergence of the results in terms of the mode number. Difference of the functions $a(t)$ and $w(x,t)$ computed with 8 and 20 modes are small. Only results with 20 modes are shown in the following.

The dimensional radius, $a(t)$, of the wetted part of the elastic cone for case 1 is shown in figure 6(a) for $\delta =0$ (circular elastic plate) and $\delta =965.3$ (conical shell with deadrise angle of $6^\circ$). It is seen that the conical shell is less flexible than the circular plate, and the radius of the wetted part of the conical shell is close to the radius for the rigid cone, $a(t)=4Vt/({\rm \pi} \sin \beta )$, shown by dashed line.

Figure 6. Comparison of the results within the Scolan model (blue lines) and the present model of conical shell (red lines): (a) dimensional radius of the wetted area $a(t)$, dashed line is for rigid cone; (b–d) coefficients $b_n(t)$ in the series (21).

The dimensionless coefficients $b_1(t)$, $b_2(t)$, $b_3(t)$ in the series (3.5) for the normal displacements of the shell elements are shown in figure 6(b,d) correspondingly for $\delta =0$ (blue lines) and for $\delta =965.3$ (red lines). It is seen that the coefficient $b_1(t)$ of the lowest mode is larger for $\delta =0$, but other coefficients with $\delta =0$ are smaller than those with $\delta \ne 0$. Therefore, the total deflection of the conical shell is smaller than in the model by Scolan (Reference Scolan2004). However, the contributions of the dry modes starting from the second one in the model by Scolan are higher than in the present model of conical shells.

Dimensional normal displacements of the cone at different time instants are shown in figure 7 for Scolan model, see figure 7(a), and the present model, figure 7(b). The scale of the normal displacement is $R\sin \beta =1.338\times 10^{-2}$ m for case 1, see (2.10a–d). The conical shell is much less deformed at the early stage. The maximum deflection of the shell is more than twice smaller than the deflection of an equivalent circular plate.

Figure 7. Normal dimensional displacement $w(x,t)$ of the circular plate (a) and the conical shell (b) for case 1 during the impact stage at time instants when $a_j/R=0.1 \cdot j$, $1\le j\le 10$. Displacement of the wetted parts of the structure are shown in blue and displacements of the dry parts are shown in red.

Calculations of the shell deflection in case 2 are performed with 20 modes and step of integration $da=10^{-5}$ and with 8 modes and step of integration $da=10^{-4}$ by Runge–Kutta fourth-order method. The results with 20 modes and with 8 modes are very close. In the following, the normal displacements with 20 modes are shown for the impact speed 6 m s$^{-1}$ (see figure 8a) and 12 m s$^{-1}$ (see figure 8b). It is seen that double increase of the impact speed results in the increase of the maximum shell deflection more than four times.

Figure 8. Normal dimensional displacement $w(x,t)$ of the conical shell for case 2 and impact speed $V=6$ m s$^{-1}$ (a) and $V=12$ m s$^{-1}$ (b) during the impact stage at time instants when $a_j/R=0.1 \cdot j$, $1\le j\le 10$. Displacement of the wetted parts of the structure are shown in blue and displacements of the dry parts are shown in red.

There are two strain components for a conical shell, $\varepsilon _{xx}(x,t)$ and $\varepsilon _{\theta \theta }(x,t)$, as shown in figure 3 and (2.19a,b). The distributions of the membrane, $\varepsilon ^{(m)}_{xx}(x,t)$ and $\varepsilon ^{(m)}_{\theta \theta }(x,t)$, and bending, $\varepsilon ^{(b)}_{xx}(x,t)$, $\varepsilon ^{(b)}_{\theta \theta }(x,t)$, strains, see (2.20a,b), along the cone at different time instants are shown in figures 9(a) and 9(b) for case 2 with impact speed 12 m s$^{-1}$. Strain are given in microstrains ($\mu s$). We may conclude that the bending strains are greater that the membrane strains in the radial direction, but the membrane strains are higher than the bending strains in the azimuthal direction. All strain components are of about the same order. The maximum strains are achieved at the edge of the cone because of bending. The strain distributions on the inner and outer surfaces of the cone are shown in figures 9(c) and 9(d). Note that the strains are singular at the tip of the cone, see the discussions in the last paragraph of § 2.

Figure 9. Distributions of the strain components at the three time instants corresponding to the radius of the contact region $a(t)=0.3$ (curve 1), $a(t)=0.6$ (curve 2) and $a(t)=0.9$ (curve 3): (a) radial membrane and bending strain components; (b) azimuthal membrane and bending strain components; radial (c) and azimuthal (d) strains along the inner $(-)$ and outer $(+)$ surfaces of the cone, $V=12$ m s$^{-1}$. The strains (in microstrains) are shown for case 2.

Strain components at two points of conical shell, $x=0.5$ and $x=0.75$, are shown as functions of the dimensionless time $\tilde {t}$ in figure 10. This figure indicates that all components of the strain should be evaluated during both, the impact and penetration stages to find the maximum strain at a certain point of the conical shell. The strain are unbounded at the tip of the cone, $x=0$. However, the elastic displacement near the tip are small and a vicinity of the tip can be treated as rigid. A better way of dealing with the strain singularity is to consider a frustum of a cone, where part near the vertex is rigid and elastic shell corresponds to $R_m< x< R$. Elastic shell is clamped to the rigid part near the vertex at $x=R_m$.

Figure 10. Radial and azimuthal strains (in microstrains) as functions of dimensionless time at dimensionless distances $x=0.5$ and $x=0.75$ from the cone tip for case 2 with impact speed 12 m s$^{-1}$. Bending and membrane components $x=0.5$ (a) and $x=0.75$ (b). Strains at the inner $(-)$ and outer $(+)$ surface of the cone, $x=0.5$ (c) and $x=0.75$ (d). Time scale is 14.5 ms.

4.3. Hydrodynamic pressure

It was discussed in § 3.3 that the hydrodynamic loads acting on the conical shell during water impact are difficult to compute and measure. However, the loads should be estimated to guarantee that they do not drop to the level where the liquid starts cavitating. In the case of very low pressures, the hydrodynamic model should be changed to include cavitation effects.

The dimensionless pressure distributions shown in figure 11 for four time instants with $a=0.2$, $0.6$, 0.8 and 1 are calculated by series (3.33) (blue lines) and by the series with improved convergence (3.41) (red lines) for Case 2 with 20 modes and speed of impact 1 m s$^{-1}$. The series (3.33) gives the pressure in the contact region, $0< r< a$, as well as in the dry part of the cone, where $a< r<1$ and $p(r,0,t)=0$. The series (3.33) does not reproduce the square-root singularity of the pressure at $r=a(t)$. The pressure distributions provided by formula (3.41) with improved convergence of the series are smooth, but some oscillations of the pressure occur due to elastic vibration of the shell. Note that the modes $\psi _n(r)$ in (3.5) are such that $\psi _n(0)=\psi _n(1)=0$, see (3.1a,b). Correspondingly, the series (3.33) incorrectly gives $p(0,0,t)=p(1,0,t)=0$ for an any finite number of retained terms.

Figure 11. Hydrodynamic pressure distributions in case 2 with entry speed 1 m s$^{-1}$, calculated by series (3.33) (blue lines) and by the series with improved convergence (3.41) (red lines) at time instants, when $a=0.2$ (a), $a=0.6$ (b), $a=0.8$ (c) and $a=1$ (d), in dimensionless variables.

The pressure distributions for case 2 and impact speed 6 m s$^{-1}$ are shown in figure 12(a) in the dimensionless variables at five time instants, which corresponds to $a=0.2$, 0.4, 0.6, 0.8 and 1. There are four types of lines in this figure. The pressures by formulae (3.33) and (3.41) are shown by green and red lines correspondingly. The black lines are for the pressures acting on the equivalent rigid cone. The blue lines show the differences between the elastic and rigid cone pressures. It is seen that the elastic and rigid pressures are different only at the end of the impact stage. Within the rigid cone model, the pressure $p^{(R)}(r,0,t)$ is given by (3.43) where the radius $a(t)$ is calculated without account for cone elasticity. The blue lines show the differences $p-p^{(R)}$ between the elastic and rigid pressures for the same values of the radius $a$. It is interesting that the elastic pressure can be higher than the rigid pressure in some places.

Figure 12. (a) Dimensionless pressure distributions in case 2 with entry speed 6 m s$^{-1}$ calculated by series (3.33) (green lines), by formula (3.41) (red lines) and without account for cone elasticity (black lines). The difference between elastic and rigid pressure are shown by blue lines. (b) Hydrodynamic pressure given by the series (41) at the end of the impact stage with 8 (magenta), 15 (black) and 20 (blue) retained modes compared with the pressure provided by (3.41) with 20 modes. (c) The difference between the elastic (3.41) and rigid (3.43) pressures, calculated with 8 (magenta), 15 (black) and 20 (blue) modes.

The convergence of the pressure given by (3.33) to the pressure provided by (3.41) with increase of the number of retained modes is demonstrated in figure 12(b) for the same conditions as in figure 12(a), but only at the end of the impact stage when $a(t)=1$. The pressure distributions by the series (3.33) with 8 modes (magenta), 15 modes (black) and 20 modes (blue) converge to the pressure predicted by (3.41) (red line) but very slowly.

The differences between the elastic (3.41) and rigid (3.43) pressures $p-p^{(R)}$, which are shown in figure 12(a) by blue lines, are shown now with 8 (magenta), 15 modes (black) and 20 modes (blue) modes retained in (3.41) in figure 12(c) for 5 time instants as in figure 12(a). This figure demonstrates fast convergence of the series (3.41) with increase of the retained modes. It is seen that the elastic pressure (3.41) is smaller then the rigid pressure (3.43) (the difference is negative) for $a=0.2$, 0.4 and 0.6, but at the end of the impact stage elastic pressure is higher than the rigid pressure. This effect can be caused by deflection of the elastic cone, when the local deadrise angle at $x=a(t)$ increases initially decreasing the hydrodynamic pressure, but then decreases at the end of the impact stage increasing the impact pressure.

4.4. Convergence of the numerical solution

Convergence issues with respect to the number of the retained modes $N$ were addressed qualitatively in §§ 4.2 and 4.3. In the present section, the convergence issues are addressed quantitatively for local quantities, such as deflection, strain and pressure distributions along the wetted part of the conical shell, at a certain time instant. All results in the following are given for case 2 with impact speed $V=6$ m s$^{-1}$ for the normal displacements of the shell elements and hydrodynamic pressures, and $V=12$ m s$^{-1}$ for both radial and azimuthal strains on the outer surface of the shell. The deflection, strain and pressure distributions are compared at the time instant $t^*$, when the dimensionless radius of wetted part of the elastic cone is equal to 0.9, $a(t^*)=0.9$.

The relative differences in percentage of the normal displacements of shell elements computed with $N$ retained modes, $w_N(x,t^*)$, and 20 modes, $w_{20}(x,t^*)$, at time $t^*$,

(4.3)\begin{equation} {{\rm \Delta}^* w(x,t^*)}=\frac{w_N(x,t^*)-w_{20}(x,t^*)}{\max_{0\le x\le 1} (w_{20}(x,t^*))} \cdot 100\,\%, \end{equation}

are shown in figure 13(a) for $N=5$, 10 and 15. The deflection $w_{20}(x,t^*)$ is shown in figure 8(a) by the line with absolute maximum, which is achieved at $x=0.72$. It is seen that relative difference between $w_{10}(x,t^*)$ and $w_{20}(x,t^*)$ is less than 0.5 % (see the magenta line). The convergence of normal displacements $w_{N}(x,t^*)$ with increase of the number of the retained modes $N$ is clear from figure 13(a).

Figure 13. Relative differences of the normal displacement (4.3) (a), radial strains (4.4) (b), azimuthal strains (4.5) (c) and impact loads (4.6) (d) calculated with $N=5$ (blue lines), $N=10$ (magenta lines) and $N=15$ (black lines) with respect to the results, calculated with 20 modes retained in the corresponding series, at time instant $t_*$, when 90 % of the cone generator is wetted.

The relative differences in percentage of the strain $\varepsilon ^+_{xx}(x,t)$ along the cone generator and the strain $\varepsilon ^+_{\theta \theta }(x,t)$ in the angular azimuthal direction (see figure 3) computed for the outer surface of the conical shell with $N$ retained modes, $\varepsilon ^+_{N,xx}(x,t)$ and $\varepsilon ^+_{N,\theta \theta }(x,t)$, at time instant $t_*$

(4.4)\begin{gather} {{\rm \Delta}^*\varepsilon}^{+}_{xx}(x,t^*)=\frac{\varepsilon^+_{N,xx}(x,t^*)- \varepsilon^+_{20,xx}(x,t^*)}{\varepsilon^+_{20,xx}(1,t^*)} \cdot 100\,\%, \end{gather}

(4.5)\begin{gather}{{\rm \Delta}^*\varepsilon}^+_{\theta\theta}(x,t^*)=\frac{\varepsilon^+_{N,\theta\theta}(x,t^*)-\varepsilon^+_{20,\theta\theta}(x,t^*)} {\varepsilon^+_{20,\theta\theta}(0.66,t^*)} \cdot 100\,\%, \end{gather}

are shown in figures 13(b) and 13(c) correspondingly for $N=5$, 10 and 15. Note that strains are unbounded at $x=0$ and, therefore, convergence of the strains should be considered out of a vicinity of the cone vertex. The denominators $\varepsilon ^+_{20,xx}(1,t^*)$ and $\varepsilon ^+_{20,\theta \theta }(0.66,t^*)$ in (4.4) and (4.5) are the extreme of the corresponding strains outside the vicinities of the vertex. It is seen that relative differences between $\varepsilon ^+_{20,xx}(x,t^*)$ and $\varepsilon ^+_{15,xx}(x,t^*)$, and between $\varepsilon ^+_{20,\theta \theta }(x,t^*)$ and $\varepsilon ^+_{15,\theta \theta }(x,t^*)$, are less than 2 % for $0.1< x<1$.

It is well known that convergence of the strains with respect to the number of retained modes is slower than convergence of the deflections, because strains are computed as second derivatives of the deflections with respect to the radial coordinate $x$. The terms in the series (3.5) for the normal displacement $w(x,t)$ are at the order $O(b_n\psi _n)=O(b_n)$ because the normalised modes (3.3) are of order $O(1)$. The second derivatives $\psi ''_n(r)$ are of order $O(n^2)$ for $0< r<1$. Therefore, the series for the strains are with terms of order $O(n^2b_n(t))$ as $n\to \infty$. From this analysis, we may conclude that the corresponding series for hydrodynamic pressure, which follows from the shell equations (2.4) and (2.5) and series (3.6) are with terms of order $O(n^4 b_n(t))$ as $n\to \infty$. The convergence of such series for the hydrodynamic pressure to the pressure provided by the series with improved convergence (3.41) is shown in figure 12(b), which demonstrates that the convergence is not acceptable for practical estimates of impact loads.

The relative differences in percentage of the pressure given by (3.41) with $N$ retained modes in (3.41) and (3.40), $p_N(x,t^*)$, and 20 modes, $p_{20}(x,t^*)$,

(4.6)\begin{equation} {{\rm \Delta}^*p(x,t^*)}=\frac{p_N(x,t^*)-p_{20}(x,t^*)}{p_{20}(x,t^*)} \cdot 100\,\%, \end{equation}

are shown in figure 13(d) for $N=5$, 10 and 15 at time instant $t^*$ when $a(t^*)=0.9$. It is seen that the relative difference between $p_{15}(x,t^*)$ and $p_{20}(x,t^*)$ is less than 0.5 %. This result confirms that the accuracy of the formula (3.41) for the impact loads is comparable with the accuracy of the series for the shell displacements.