Bubble-driven films, membranes and detachment

Source: supplied generic textbook, Chapter 12. The complete bilingual chapter follows, with its derivations and qualifications. Textbook source (Markdown) · Original Markdown.

来源：所给通用教材第 12。下方完整收录双语章节及推导和限定。教材原文（Markdown） · 原始 Markdown。

12. Bubble-driven films, membranes, and detachment 12.1 A liquid pressure is only one part of structural loading 12.2 Plate and membrane limits 12.3 Modal projection shows why load shape matters 12.4 Added mass without double counting 12.5 Detachment is a fracture problem 12.6 A useful hierarchy of questions

12. Bubble-driven films, membranes, and detachment

气泡驱动薄膜、膜结构与脱离

12.1 A liquid pressure is only one part of structural loading

液体压力只是结构载荷的一部分

A deformable solid experiences the fluid traction $\mathbf t_f=\boldsymbol\sigma_f\mathbf n$ , containing normal pressure and viscous stress. At a bonded fluid–solid interface, velocity compatibility and traction continuity close the coupling. A scalar bubble pressure can be used as a reduced load only after explaining how it is distributed over the solid. [S362, S377]

可变形固体承受流体牵引 $\mathbf t_f=\boldsymbol\sigma_f\mathbf n$ ，其中包含法向压力和黏性应力。在无滑移流固界面上，速度相容与牵引连续共同闭合耦合问题。只有解释清楚压力如何分布于固体表面后，才能将单一气泡压力用作降阶载荷。[S362, S377]

The wall-shear study [S362] links deformation measurements and fluid–structure simulations. Its significance for mechanics is that maximum tangential loading need not coincide with maximum normal displacement or lie directly under the bubble center. An adhesion problem may therefore require both normal and shear loading.

壁面剪切研究[S362]将变形测量与流固耦合仿真结合起来。其力学意义在于，最大切向载荷不一定与最大法向位移重合，也不一定位于气泡中心正下方。因此，黏附问题可能需要同时考虑法向与剪切加载。

12.2 Plate and membrane limits

板与膜的极限

For a thin, initially flat, isotropic layer with small slopes and approximately linear response, a useful transverse equation is

对于薄、初始平直、各向同性，且斜率小、响应近似线性的层状结构，一个实用横向方程为

m_A w_{tt}+c_Aw_t+D\nabla^4w-N_0\nabla^2w=\Delta p(\mathbf x,t), \qquad D=\frac{Eh^3}{12(1-\nu_s^2)}. \tag{12.1}

12.2 Plate and membrane limits / 板与膜的极限; m_A: moving mass per unit area; w_tt: transverse film displacement; c_A: sound speed / heat capacity; w_t: transverse film displacement; D: plate bending rigidity; w: transverse film displacement; N_0: event count / membrane tension; Δ: change / pressure difference; p: pressure at the indicated location; x: position / dimensionless radius; t: physical time; ν_s: Poisson ratio

Here $m_A$ is mass per area, $c_A$ a distributed viscous damping coefficient, $D$ bending rigidity, and $N_0$ isotropic pretension per length. The solid Poisson ratio is written $\nu_s$ to distinguish it from liquid kinematic viscosity $\nu$ . A membrane limit neglects bending; a plate limit may neglect pretension. Large deflection generates additional in-plane stress and generally needs geometrically nonlinear theory.

这里 $m_A$ 为面密度， $c_A$ 为分布黏性阻尼系数， $D$ 为弯曲刚度， $N_0$ 为单位长度上的各向同性预张力。固体泊松比记为 $\nu_s$ ，以区别于液体运动黏度 $\nu$ 。膜极限忽略弯曲，板极限则可能忽略预张力。大挠度会产生附加面内应力，通常需要几何非线性理论。

Boundary conditions matter as much as material constants. A clamped circular edge enforces both zero displacement and zero slope in a thin-plate model. A simply supported edge imposes a different moment condition. A flexible layer bonded to a payload is a composite structure; adding payload mass without accounting for its stiffness is a modeling choice that must be justified.

边界条件的重要性不亚于材料常数。在薄板模型中，圆形固支边缘同时要求位移和转角为零；简支边缘则具有不同弯矩条件。柔性层与负载薄膜粘接后成为复合结构；仅增加负载质量而忽略其刚度，是一项需要论证的建模选择。

12.3 Modal projection shows why load shape matters

模态投影解释载荷分布的重要性

Write $w(\mathbf x,t)=\sum_nq_n(t)\phi_n(\mathbf x)$ . With a chosen normalization, projection gives

写成 $w(\mathbf x,t)=\sum_nq_n(t)\phi_n(\mathbf x)$ ，在选定归一化条件下，模态投影得到

M_n\ddot q_n+C_n\dot q_n+K_nq_n=F_n(t),\qquad F_n(t)=\int_A\Delta p(\mathbf x,t)\phi_n(\mathbf x)\,dA. \tag{12.2}

The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.

图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。

The modal impulse is $I_n=\int F_n(t)dt$ , not merely the pressure impulse at the center. Two load distributions can have the same total force impulse yet excite different modes, because the weighting functions $\phi_n$ differ. This is the structural counterpart of the spatial pressure-impulse argument in Chapter 9.

模态冲量为 $I_n=\int F_n(t)dt$ ，而不只是中心点的压力冲量。即使两个载荷分布具有相同总力冲量，也可能激发不同模态，因为权函数 $\phi_n$ 不同。这是第9章空间压力冲量思想在结构动力学中的对应形式。

For a sufficiently short pulse relative to $1/\omega_n$ , initial modal displacement changes little and $\Delta\dot q_n\simeq I_n/M_n$ . The ensuing undamped amplitude is approximately $I_n/(M_n\omega_n)$ . For a rectangular force pulse of duration $\tau_p$ and the same total impulse, the exact post-pulse free amplitude is

对于相对于 $1/\omega_n$ 足够短的脉冲，初始模态位移变化很小，且 $\Delta\dot q_n\simeq I_n/M_n$ ，后续无阻尼振幅约为 $I_n/(M_n\omega_n)$ 。对于总冲量相同、持续时间为 $\tau_p$ 的矩形力脉冲，脉冲结束后的精确自由振幅为

A_n=\frac{|I_n|}{M_n\omega_n} \left|\frac{2\sin(\omega_n\tau_p/2)}{\omega_n\tau_p}\right|. \tag{12.3}

12.3 Modal projection shows why load shape matters / 模态投影解释载荷分布的重要性; A_n: area / volume-acceleration coefficient; I_n: pressure or modal impulse; M_n: generalized mass / molar mass; ω_n: structural modal angular frequency; τ_p: duration / dimensionless time / dummy time

Thus equal impulse does not guarantee equal final vibration when the load is not short. The force may do positive work during one part of the motion and negative work during another. A high pressure peak with poor temporal coupling may be less effective than a lower, better-timed pulse.

因此，当加载不够短时，相同冲量并不能保证相同最终振动。力可能在运动的一部分阶段做正功，在另一部分阶段做负功。一个时间耦合不佳的高压峰值，可能不如较低但时序更合适的脉冲有效。

A single undamped structural mode under rectangular pulses with equal impulse and different durations. / 单个无阻尼结构模态在相同冲量、不同持续时间矩形脉冲下的响应。 — A single undamped structural mode under rectangular pulses with equal impulse and different durations.
单个无阻尼结构模态在相同冲量、不同持续时间矩形脉冲下的响应。

12.4 Added mass without double counting

不重复计算附加质量

An oscillating film accelerates adjacent liquid, which can lower natural frequencies relative to vacuum values. In a reduced structural model this may be represented by a frequency- and geometry-dependent added mass. In a fully coupled fluid–structure simulation, that liquid inertia is already present in the fluid domain. Adding the same hydrodynamic mass to the structural density would count it twice.

振动薄膜会加速邻近液体，因此其固有频率可能低于真空值。在降阶结构模型中，可用与频率和几何相关的附加质量表示这一影响；而在完整流固耦合仿真中，液体惯性已由流体域体现。若再把相同水动力质量加到结构密度中，就会重复计算。

The ratio $\Theta_n=\omega_n t_b$ , with $t_b$ a bubble-loading time, helps distinguish impulsive, resonant-scale, and slowly varying forcing. It is not sufficient by itself: the spatial mode overlap and damping still matter.

比值 $\Theta_n=\omega_n t_b$ 可用于区分冲量式、与固有振动同量级，以及缓变加载，其中 $t_b$ 为气泡加载时间。但仅有该比值还不够，空间模态匹配和阻尼仍然重要。

12.5 Detachment is a fracture problem

脱离是断裂问题

A film that deflects is not necessarily detached. Interfacial separation requires a failure law. In a fracture description, the energy release rate $G$ is compared with an appropriate critical value $G_c$ , potentially depending on mode mix, rate, temperature, and surface condition. In a cohesive-zone description, traction depends on local separation and loading history. The hydrogel-stamp paper [S72] shows why changing contact geometry and adhesion can be central to successful transfer.

薄膜发生挠曲，不代表已经脱离。界面分离需要失效定律。在断裂力学描述中，能量释放率 $G$ 应与适当的临界值 $G_c$ 比较，而 $G_c$ 可能依赖混合模态、速率、温度和表面状态。在内聚区描述中，牵引取决于局部分离量与加载历史。水凝胶印章论文[S72]表明，改变接触几何与黏附状态可能是成功转印的关键。

For an illustrative triangular mode-I cohesive law with peak traction $T_c$ and final separation $\delta_f$ , the area under the traction–separation curve is $G_c=T_c\delta_f/2$ . A large penalty stiffness before damage should enforce the intended initial bonding without making the numerical system needlessly ill-conditioned. Mixed-mode or rate-dependent separation requires more than this elementary law.

对于示意性的三角形I型内聚定律，若峰值牵引为 $T_c$ 、最终分离量为 $\delta_f$ ，则牵引—分离曲线下面积为 $G_c=T_c\delta_f/2$ 。损伤前的惩罚刚度应足以实现预期初始粘接，同时避免使数值系统不必要地病态。混合模态或速率相关分离需要超出这一基础定律的描述。

A very rough areal kinetic-energy comparison is $\tfrac12m_Av^2\gtrsim G_c$ , giving $v\gtrsim\sqrt{2G_c/m_A}$ . This is only a necessary energy-scale argument under idealized allocation of energy to separation; it is not a sufficient release criterion. Energy can remain in bending, fluid motion, heating, or rebound, and fracture initiation depends on geometry and defects.

非常粗略的面动能比较为 $\tfrac12m_Av^2\gtrsim G_c$ ，从而得到 $v\gtrsim\sqrt{2G_c/m_A}$ 。这只是理想化能量全部用于分离时的必要能量尺度论证，并不是充分释放判据。能量可能继续储存在弯曲、流体运动、热量或回弹中，而断裂起始还依赖几何与缺陷。

12.6 A useful hierarchy of questions

有效的问题层次

First ask whether the bubble can move the layer. Then ask whether the resulting loading can initiate and propagate separation. Finally ask whether the released object reaches and remains on the receiver without damage. Solving these as separate stages before coupling them prevents an apparently impressive simulation from hiding an untested adhesion assumption.

首先判断气泡能否推动结构层，然后判断形成的载荷能否启动并推进分离，最后判断释放对象能否无损到达并保留在接收面上。在完全耦合之前先分别解决这些阶段，可以避免看似复杂的仿真掩盖未经检验的黏附假设。

Integrated calculation and model check

综合计算与模型检查

Use the declared parameter set in each part. The values are teaching scenarios unless specifically identified as inherited project inputs. Each reference panel follows its question.

使用每小题明确给出的参数。除明确标为沿用项目输入外，数值均为教学情景。各参考解答紧随题目。

A clamped plate has E = 3 GPa, ν = 0.35 and h = 5 µm. Compute D. For a = 50 µm and uniform Δp = 10 kPa estimate the bending screening ratio Δp a⁴/(Dh).
夹持板 E = 3 GPa、ν = 0.35、h = 5 µm，计算 D。a = 50 µm、均匀 Δp = 10 kPa 时，估算弯曲筛查比 Δp a⁴/(Dh)。
Reference answer / 参考答案
$D=Eh^3/[12(1-\nu^2)],\qquad \Pi_b=\Delta p\,a^4/(Dh)$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
D = 3.561×10⁻⁸ N m, Πb = 0.351. This is a scaling ratio; geometry factors, membrane tension, added mass and large-deflection stretching determine the actual displacement. It is not an exact clamped-plate deflection.
D = 3.561×10⁻⁸ N m，Πb = 0.351。这是尺度比；几何系数、膜张力、附加质量和大挠度拉伸决定真实位移。它不是夹持板挠度的精确解。
Two rectangular modal loads have the same impulse In. One lasts much less than a modal period; one lasts exactly one undamped period. Compare the residual oscillation amplitudes.
两种矩形模态载荷具有相同冲量 In；一个远短于模态周期，另一个恰好持续一个无阻尼周期。比较卸载后残余振幅。
Reference answer / 参考答案
$q_{\mathrm{free}}=\frac{|I_n|}{M_n\omega_n}\left|\frac{2\sin(\omega_n\tau_p/2)}{\omega_n\tau_p}\right|$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
For a very short pulse the factor approaches 1; at τp = 2π/ωn it is 0. Equal net impulse does not imply equal peak displacement or residual energy. Motion during the long pulse need not be zero.
极短脉冲时系数趋于 1；τp = 2π/ωn 时为 0。总冲量相同不意味着峰值位移或残余能量相同。长脉冲期间运动并不必然为零。
Take moving areal mass mA = 0.020 kg/m² and Gc = 0.10 J/m². Compute nominal separation velocity and impulse scales. State why they do not certify release.
运动面密度 mA = 0.020 kg/m²、Gc = 0.10 J/m²，计算名义分离速度与冲量尺度，并说明它们为何不能证明释放。
Reference answer / 参考答案
$\tfrac12m_Av^2=G_c,\qquad I_{p,\mathrm{nom}}=\sqrt{2m_AG_c}$
$Bubble-driven films, membranes and detachment · original governing laws; m_A: moving mass per unit area; v: velocity / squared wall speed as defined; G_c: fracture work per release area; I_p,nom: pressure or modal impulse$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
v = 3.162 m/s, Ip,nom = 0.06325 Pa s. These are necessary energy screens under full useful conversion; localized traction, crack propagation and losses can prevent separation. A coupled fluid solve already includes liquid inertia, so adding the same full added mass again double counts it.
v = 3.162 m/s，Ip,nom = 0.06325 Pa s。这些是假设全部有效转化时的必要能量筛查；局部牵引、裂纹传播及损耗可能阻止分离。耦合流体求解已包含液体惯性，再加相同完整附加质量会重复计算。

Brief conceptual synthesis

简短概念综合

After using the reference panels, give a short explanation of the physical mechanism, the strongest assumption, and one observation that could falsify this chapter’s model. Preparation of later chapters does not require waiting for that reply.

核对参考解答后，简述物理机制、最强假设，以及一个可证伪本章模型的观测。后续章节编写无需等待该回答。