Structural motion and interfacial release

Source: supplied project-theory book, Stage 4. The complete bilingual chapter follows, with its derivations and qualifications. Textbook source (Markdown) · Original Markdown.

来源：所给项目理论教材阶段 4。下方完整收录双语章节及推导和限定。教材原文（Markdown） · 原始 Markdown。

Keep the source’s evidence states: optical normalization is an accounting check; activation is uncalibrated; R(t) is prescribed; near-wall loading and cohesive release remain conditional.

保留来源证据状态：光学归一化属于核算检查；激活尚未标定；R(t) 为规定输入；近壁加载及黏聚释放仍有条件。

4. Structural motion and interfacial release 4.1 Reducing the laminate to one generalized coordinate Modal mass Modal stiffness Generalized load 4.2 Why load timing matters 4.3 Why equal total impulse can also fail spatially 4.4 Deriving first-arrival wave transmission 4.5 From transmitted loading to cohesive failure Cohesive-zone length scale 4.6 Deriving the impulse and energy availability ratios

4. Structural motion and interfacial release

结构运动与界面释放

From pressure to structural motion and interfacial release. Corresponding to Theory 04, source page 4.

从压力到结构运动，再到界面释放。对应 Theory 04，原始材料第4页。

The fourth page contains two complementary mechanical reductions: a flexural mode describing laminate motion, and a first-arrival wave calculation describing through-thickness transmission. Neither, by itself, supplies a resolved cohesive-interface traction history. [S1, p. 4]

第4页包含两种互补的力学简化：用弯曲模态描述层合结构运动，以及用首波计算描述厚度方向的传输。这两种简化中的任何一种，都不能单独给出经过空间解析的黏聚界面牵引历程。[S1，第4页]

4.1 Reducing the laminate to one generalized coordinate

将层合结构简化为一个广义坐标

Approximate the transverse displacement as

将横向位移近似为

w(s,t)=\phi(s)q(t),

4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标; w: transverse film displacement; s: in-plane radial coordinate; t: physical time; φ: assumed spatial mode shape; q: generalized mode displacement

where $\phi(s)$ is an assumed spatial shape and $q(t)$ its amplitude.

其中， $\phi(s)$ 是假定的空间振型， $q(t)$ 是其幅值。

The printed modal coefficients can be reconstructed using the clamped-disk shape

采用以下固支圆盘振型，可以重构原始材料中的模态系数：

\boxed{\phi(s)=\left(1-\frac{s^2}{a^2}\right)^2},\qquad 0\le s\le a.

4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标; φ: assumed spatial mode shape; s: in-plane radial coordinate; a: droplet radius / film span / acceleration

It satisfies

它满足

\phi(a)=0,\qquad\phi'(a)=0,\qquad\phi(0)=1.

4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标; φ: assumed spatial mode shape; a: droplet radius / film span / acceleration; φ': assumed spatial mode shape

Thus the edge is clamped and $q(t)$ is the centre displacement. This is an admissible assumed shape, not a claim that the exact laminate eigenmode has been measured.

因此，边缘为固支条件， $q(t)$ 就是中心位移。这是一个满足边界条件的假定振型，并不意味着已经测得层合结构的精确固有模态。

模态质量

Let $m'$ be laminate mass per unit area. The kinetic energy is

以 $m'$ 表示层合结构的单位面积质量。动能为

T=\frac12\int_A m'\dot w^2\,dA=\frac12\dot q^2\int_A m'\phi^2\,dA.

Modal mass / 模态质量; T: temperature / kinetic energy as defined; m': mass or areal mass; ẇ: transverse film displacement; A: area / volume-acceleration coefficient; q̇: generalized mode displacement; φ: assumed spatial mode shape

Therefore,

因此

M_q=2\pi m'\int_0^a\phi^2s\,ds.

Modal mass / 模态质量; M_q: generalized modal mass; π: circle geometry constant; m': mass or areal mass; a: droplet radius / film span / acceleration; φ: assumed spatial mode shape; s: in-plane radial coordinate

Since

由于

\int_0^a\left(1-\frac{s^2}{a^2}\right)^4s\,ds=\frac{a^2}{10},

Modal mass / 模态质量; a: droplet radius / film span / acceleration; s: in-plane radial coordinate

we obtain

可得

\boxed{M_q=\frac{\pi m'a^2}{5}}.

Modal mass / 模态质量; M_q: generalized modal mass; π: circle geometry constant; m': mass or areal mass; a: droplet radius / film span / acceleration

模态刚度

For an equivalent isotropic thin-plate reduction with bending rigidity $D$ ,

对于弯曲刚度为 $D$ 的等效各向同性薄板简化模型，

U=\frac{D}{2}\int_A\left(\kappa_r^2+\kappa_\theta^2+2\nu\kappa_r\kappa_\theta\right)dA,

Modal stiffness / 模态刚度; U: velocity / velocity scale; D: plate bending rigidity; κ_r: gas polytropic exponent / curvature; κ_θ: gas polytropic exponent / curvature; ν: Poisson ratio; A: area / volume-acceleration coefficient

where

其中

\kappa_r=-q\phi'',\qquad\kappa_\theta=-q\frac{\phi'}s.

Modal stiffness / 模态刚度; κ_r: gas polytropic exponent / curvature; q: generalized mode displacement; φ'': assumed spatial mode shape; κ_θ: gas polytropic exponent / curvature; φ': assumed spatial mode shape; s: in-plane radial coordinate

Matching $U=\tfrac12K_qq^2$ gives

与 $U=\tfrac12K_qq^2$ 对比，得到

K_q=2\pi D\int_0^a\left[(\phi'')^2+\left(\frac{\phi'}s\right)^2+2\nu\phi''\frac{\phi'}s\right]s\,ds.

Modal stiffness / 模态刚度; K_q: generalized modal stiffness; π: circle geometry constant; D: plate bending rigidity; a: droplet radius / film span / acceleration; φ'': assumed spatial mode shape; φ': assumed spatial mode shape; s: in-plane radial coordinate; ν: Poisson ratio

For the selected shape,

对于选定的振型，

\int_0^a[\cdots]s\,ds=\frac{32}{3a^2},

Modal stiffness / 模态刚度; a: droplet radius / film span / acceleration; s: in-plane radial coordinate

所以

\boxed{K_q=\frac{64\pi D}{3a^2}}.

The actual laminate rigidity $D$ still requires its layer constitutive properties and appropriate laminate assumptions.

实际层合结构的刚度 $D$ 仍需根据各层本构参数与适当的层合理论假设确定。

Generalized load

广义载荷

Virtual work is

虚功为

\delta W=\int_A p(s,t)\,\delta w\,dA.

Generalized load / 广义载荷; δ: separation / perturbation / penetration depth; W: work / potential energy; p: pressure at the indicated location; s: in-plane radial coordinate; t: physical time; w: film displacement / laser waist; A: area / volume-acceleration coefficient

Since $\delta w=\phi\,\delta q$ ,

由于 $\delta w=\phi\,\delta q$ ，

\delta W=\left[\int_A p(s,t)\phi(s)\,dA\right]\delta q.

Hence,

因此

\boxed{Q(t)=2\pi\int_0^a p(s,t)\phi(s)s\,ds}.

Generalized load / 广义载荷; Q: heat or generalized load as defined; t: physical time; π: circle geometry constant; a: droplet radius / film span / acceleration; p: pressure at the indicated location; s: in-plane radial coordinate; φ: velocity potential / mode shape

Here $Q(t)$ is a generalized force, not the optical heating source $Q'''$ .

这里的 $Q(t)$ 是广义力，不是光学体热源 $Q'''$ 。

Lagrange’s equation, with modal damping $C_q$ , now gives

考虑模态阻尼 $C_q$ ，由 Lagrange 方程得到

\boxed{M_q\ddot q+C_q\dot q+K_qq=Q(t)}.

Generalized load / 广义载荷; M_q: generalized modal mass; q̈: heat flux / mode displacement; C_q: generalized modal damping; q̇: heat flux / mode displacement; K_q: generalized modal stiffness; q: heat flux / mode displacement; Q: heat or generalized load as defined; t: physical time

This is the reduced structural equation in the source. Its load is a spatial projection, not merely the pressure at the centre. [S1, p. 4]

这就是原始材料中的降阶结构方程。其载荷是压力场的空间投影，而不只是中心点的压力。[S1，第4页]

4.2 Why load timing matters

为什么载荷时序很重要

Define

定义

\omega_n=\sqrt{\frac{K_q}{M_q}},\qquad\zeta=\frac{C_q}{2\sqrt{M_qK_q}},\qquad\omega_d=\omega_n\sqrt{1-\zeta^2}.

4.2 Why load timing matters / 为什么载荷时序很重要; ω_n: structural modal angular frequency; K_q: generalized modal stiffness; M_q: generalized modal mass; ζ: modal damping ratio; C_q: generalized modal damping; ω_d: damped modal angular frequency

For zero initial conditions and an underdamped mode,

在零初始条件及欠阻尼模态下，

\boxed{q(t)=\frac{1}{M_q\omega_d}\int_0^t Q(\tau)e^{-\zeta\omega_n(t-\tau)}\sin\!\left[\omega_d(t-\tau)\right]\,d\tau}.

4.2 Why load timing matters / 为什么载荷时序很重要; q: heat flux / mode displacement; t: physical time; M_q: generalized modal mass; ω_d: damped modal angular frequency; Q: heat or generalized load as defined; τ: duration / dimensionless time / dummy time; ζ: modal damping ratio; ω_n: structural modal angular frequency

This follows by convolving the load with the oscillator’s impulse response.

这是将载荷与振子的脉冲响应进行卷积得到的结果。

The sine factor is the key. The same load increment can increase or decrease the current motion depending on when it arrives. Therefore, two histories with equal net impulse need not produce equal peak displacement. This mathematically explains the phase sensitivity discussed on source page 3. [S1, p. 3]

关键在于正弦因子。同样的载荷增量，根据到达时刻的不同，可能增强当前运动，也可能削弱当前运动。因此，净冲量相同的两条载荷历程，不一定产生相同的峰值位移。这从数学上解释了原始材料第3页讨论的相位敏感性。[S1，第3页]

4.3 Why equal total impulse can also fail spatially

为什么总冲量相同仍可能产生不同的空间响应

Write a separable pressure distribution as

将可分离的压力分布写成

p(s,t)=F(t)b(s),\qquad\int_A b(s)\,dA=1.

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。

Then $F(t)$ is total force, while

此时 $F(t)$ 为总力，而

Q(t)=F(t)\underbrace{\int_A b(s)\phi(s)\,dA}_{\Lambda}.

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。

Two distributions can have the same total force history but different modal participation factors $\Lambda$ .

两个分布可以具有相同的总力历程，却具有不同的模态参与因子 $\Lambda$ 。

For example, uniform loading over the entire disk has

例如，整个圆盘上的均匀载荷对应

b=\frac1{\pi a^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。

giving

因此

\Lambda=\frac1{\pi a^2}\int_A\phi\,dA=\frac13.

4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应; Λ: modal participation / viscous ratio; π: circle geometry constant; a: droplet radius / film span / acceleration; φ: velocity potential / mode shape; A: area / volume-acceleration coefficient

A strongly centre-concentrated load samples $\phi\approx1$ , giving a participation factor approaching one.

高度集中于中心附近的载荷作用在 $\phi\approx1$ 的区域，因此其参与因子接近1。

This is an illustrative consequence of the assumed shape, not a numerical reconstruction of the source’s bars. It explains why the top-hat, Gaussian, and raised-cosine distributions in the source produce different centre displacements despite equal total impulse. [S1, p. 4]

这是采用该假定振型后得到的示意性结论，并不是对原始材料中柱状图的数值复现。它解释了为什么原始材料中的顶帽、高斯和升余弦分布，即使总冲量相同，也会产生不同的中心位移。[S1，第4页]

4.4 Deriving first-arrival wave transmission

推导首波传输

For one-dimensional longitudinal motion in an isotropic elastic layer,

对于各向同性弹性层中的一维纵向运动，

\rho\frac{\partial^2u}{\partial t^2}=\frac{\partial\sigma_{zz}}{\partial z},\qquad\sigma_{zz}=M_L\frac{\partial u}{\partial z},

4.4 Deriving first-arrival wave transmission / 推导首波传输; ρ: liquid / layer / droplet density; u: liquid or solid particle velocity; t: physical time; σ_zz: interface tension / solid stress; z: depth / formulation predictor; M_L: generalized mass / molar mass

where the longitudinal modulus is

其中纵向模量为

M_L=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; M_L: generalized mass / molar mass; E: energy / elastic modulus as defined; ν: Poisson ratio

Combining gives a wave equation with speed

将两式结合，得到波动方程，其波速为

\boxed{c_L=\sqrt{\frac{E(1-\nu)}{\rho(1+\nu)(1-2\nu)}}}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; c_L: sound speed / heat capacity; E: energy / elastic modulus as defined; ν: Poisson ratio; ρ: liquid / layer / droplet density

The corresponding impedance is

相应的阻抗为

\boxed{Z=\rho c_L}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; Z: wave impedance; ρ: liquid / layer / droplet density; c_L: sound speed / heat capacity

For a normally incident wave at an ideal bonded interface, let $p_i,p_r,p_t$ denote incident, reflected, and transmitted compressive amplitudes. Continuity of traction and particle velocity gives

对于垂直入射到理想粘结界面的波，以 $p_i,p_r,p_t$ 分别表示入射、反射与透射的压缩波幅值。由牵引连续和质点速度连续条件可得

p_i+p_r=p_t,

4.4 Deriving first-arrival wave transmission / 推导首波传输; p_i: pressure at the indicated location; p_r: pressure at the indicated location; p_t: pressure at the indicated location

\frac{p_i-p_r}{Z_1}=\frac{p_t}{Z_2}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; p_i: pressure at the indicated location; p_r: pressure at the indicated location; Z_1: wave impedance; p_t: pressure at the indicated location; Z_2: wave impedance

Solving,

求解得到

\boxed{R_p=\frac{p_r}{p_i}=\frac{Z_2-Z_1}{Z_2+Z_1}},

4.4 Deriving first-arrival wave transmission / 推导首波传输; R_p: bubble wall radius; p_r: pressure at the indicated location; p_i: pressure at the indicated location; Z_2: wave impedance; Z_1: wave impedance

\boxed{T_p=\frac{p_t}{p_i}=\frac{2Z_2}{Z_1+Z_2}}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; T_p: temperature / kinetic energy as defined; p_t: pressure at the indicated location; p_i: pressure at the indicated location; Z_2: wave impedance; Z_1: wave impedance

Because energy flux scales as $p^2/Z$ , the energy-transmission coefficient is

由于能流按 $p^2/Z$ 缩放，能量透射系数为

\boxed{T_I=T_p^2\frac{Z_1}{Z_2}=\frac{4Z_1Z_2}{(Z_1+Z_2)^2}}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; T_I: temperature / kinetic energy as defined; T_p: temperature / kinetic energy as defined; Z_1: wave impedance; Z_2: wave impedance

Thus $T_p>1$ does not imply that transmitted energy exceeds incident energy.

因此， $T_p>1$ 并不意味着透射能量超过入射能量。

For a stack,

对于多层堆叠结构，

\boxed{t_{\mathrm{transit}}=\sum_i\frac{h_i}{c_{L,i}}}.

4.4 Deriving first-arrival wave transmission / 推导首波传输; t_transit: physical time; h_i: stand-off / layer thickness; c_L,i: sound speed / heat capacity

The source reports a one-way value of $29.9\ \mathrm{ns}$ , but its calculation retains only the forward first arrival. Later reflections, attenuation, shear transmission, and the resolved through-thickness field are not included. A short first-arrival time does not make those omitted effects irrelevant. [S1, p. 4]

原始材料报告的单程传播时间为 $29.9\ \mathrm{ns}$ ，但计算仅保留了首次正向到达的波。后续反射、衰减、剪切传输，以及完整解析的厚度方向场均未包含在内。首波到达时间很短，并不意味着这些被忽略的效应不重要。[S1，第4页]

4.5 From transmitted loading to cohesive failure

从传递载荷到黏聚失效

Interfacial release requires a relation between traction and displacement discontinuity.

界面释放需要通过牵引与位移不连续之间的关系来描述。

The source introduces a cohesive form

原始材料引入了如下黏聚形式：

\boxed{\mathbf t=(1-D_c)\mathbf K\boldsymbol\delta},

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。

where $\boldsymbol\delta$ contains opening and sliding separations, $\mathbf t$ contains the corresponding tractions, and $D_c$ is a damage variable.

其中， $\boldsymbol\delta$ 包含张开与滑移分离量， $\mathbf t$ 包含相应的牵引分量， $D_c$ 为损伤变量。

Its quadratic mixed-mode initiation condition is

其二次混合模态损伤起始条件为

\boxed{\left(\frac{\langle t_n\rangle}{T_{n0}}\right)^2+\left(\frac{t_s}{T_{s0}}\right)^2=1},

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。

with $\langle t_n\rangle=\max(t_n,0)$ under a tension-positive convention.

在拉伸为正的符号约定下， $\langle t_n\rangle=\max(t_n,0)$ 。

This condition identifies the onset of damage in the adopted model. It does not by itself specify complete separation. That requires the subsequent traction–separation evolution and fracture energy, schematically

这一条件确定的是所采用模型中的损伤起始，并不能单独确定完全分离。要描述完全分离，还需要后续的牵引–分离演化关系与断裂能，其示意表达为

G_c=\int_{\text{separation path}}\mathbf t\cdot d\boldsymbol\delta.

$4.5 From transmitted loading to cohesive failure / 从传递载荷到黏聚失效; G_c: fracture work per release area; t: physical time; δ: separation / perturbation / penetration depth$

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 distinction is analogous to strength versus toughness: reaching an initiation stress is not the same as supplying the work required to propagate separation.

这种区分类似于强度与韧度的区别：达到损伤起始应力，并不等同于提供了使分离继续扩展所需的功。

The source explicitly identifies missing normal and tangential traction histories, rate-dependent strength, mixed-mode fracture energy, and a resolved cohesive zone. [S1, p. 4]

原始材料明确指出，当前缺少法向与切向牵引历程、速率相关强度、混合模态断裂能，以及经过空间解析的黏聚区。[S1，第4页]

There is also a sign-convention issue: positive fluid pressure is compressive, whereas positive cohesive normal traction usually denotes opening tension. The transmitted pressure curve cannot simply be relabelled as cohesive traction without solving the relevant structural/interface problem.

这里还存在符号约定问题：正的流体压力表示压缩，而正的黏聚法向牵引通常表示张开拉伸。在没有求解相应结构或界面问题之前，不能简单地将透射压力曲线改名为黏聚牵引曲线。

Cohesive-zone length scale

黏聚区长度尺度

The source’s scaling

原始材料给出的尺度关系

\ell_{\mathrm{cz}}\sim\frac{E'G_c}{T_0^2}

Cohesive-zone length scale / 黏聚区长度尺度; ℓ_cz: cohesive-zone length scale; E': energy / elastic modulus as defined; G_c: fracture work per release area; T_0: temperature / kinetic energy as defined

can be understood from fracture mechanics. With

可以从断裂力学角度理解。由

K_c^2\sim E'G_c

$Cohesive-zone length scale / 黏聚区长度尺度; K_c: stiffness / collapse constant; E': energy / elastic modulus as defined; G_c: fracture work per release area$

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。

and near-tip stress scaling

以及裂纹尖端附近的应力尺度关系

\sigma\sim\frac{K_c}{\sqrt r},

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 distance at which stress reaches cohesive strength $T_0$ is

可得应力达到黏聚强度 $T_0$ 时的距离尺度

r\sim\left(\frac{K_c}{T_0}\right)^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。

Therefore,

因此

\boxed{\ell_{\mathrm{cz}}\sim\frac{E'G_c}{T_0^2}}.

The coefficient depends on the cohesive law and geometry; the scaling explains why a cohesive simulation needs a sufficiently fine mesh.

其比例系数取决于黏聚定律和几何条件；这一尺度关系说明了为什么黏聚仿真需要足够精细的网格。

4.6 Deriving the impulse and energy availability ratios

推导冲量与能量可用性比值

Let $J_A$ be the selected pressure impulse per unit area and $m'$ the moving mass per unit area. For an idealized rigid layer initially at rest,

以 $J_A$ 表示所选压力历程的单位面积冲量，以 $m'$ 表示运动部分的单位面积质量。对于初始静止的理想刚性层，

m'v=J_A.

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; m': mass or areal mass; v: velocity / squared wall speed as defined; J_A: selected areal pressure impulse

Its kinetic energy per unit area is then

其单位面积动能为

\frac{E}{A}=\frac12m'v^2=\boxed{\frac{J_A^2}{2m'}}.

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; E: energy / elastic modulus as defined; A: area / volume-acceleration coefficient; m': mass or areal mass; v: velocity / squared wall speed as defined; J_A: selected areal pressure impulse

Equating this idealized available energy to $G_c$ defines a nominal impulse scale:

令这一理想化的可用能量等于 $G_c$ ，可定义名义冲量尺度：

\boxed{J_{\mathrm{nominal}}=\sqrt{2m'G_c}}.

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; J_nominal: nucleation rate / areal impulse; m': mass or areal mass; G_c: fracture work per release area

The source therefore forms

因此，原始材料构造了

\boxed{\Pi_J=\frac{J_A}{J_{\mathrm{nominal}}}},

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; Π_J: impulse availability ratio; J_A: selected areal pressure impulse; J_nominal: nucleation rate / areal impulse

and separately,

以及另一个独立比值

\boxed{\Pi_E=\frac{E_{\mathrm{mode,max}}}{A_{\mathrm{rel}}G_c}}.

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; Π_E: energy availability ratio; E_mode,max: energy / elastic modulus as defined; A_rel: area / volume-acceleration coefficient; G_c: fracture work per release area

The reported values,

原始材料给出的数值为

\Pi_J=0.565,\qquad\Pi_E=0.767,

4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值; Π_J: impulse availability ratio; Π_E: energy availability ratio

are scalar availability comparisons. They are neither release probabilities nor local energy-release rates. They also need not obey $\Pi_E=\Pi_J^2$ , because the modal-energy calculation and rigid-layer impulse estimate are different reductions. [S1, p. 4]

它们属于标量可用性比较，既不是释放概率，也不是局部能量释放率。它们也不必满足 $\Pi_E=\Pi_J^2$ ，因为模态能量计算与刚性层冲量估计采用了不同的简化模型。[S1，第4页]

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。

An illustrative clamped disk has a = 50 µm, m′ = 0.020 kg/m² and D = 3.0×10⁻⁸ N m. Using the source trial shape, calculate Mq, Kq, fn and the modal force for uniform 10-kPa pressure.
示例夹持圆盘 a = 50 µm、m′ = 0.020 kg/m²、D = 3.0×10⁻⁸ N m。使用来源试函数，计算 Mq、Kq、fn 及均匀 10 kPa 压力的模态力。
Reference answer / 参考答案
$\phi=(1-s^2/a^2)^2,\quad M_q=\pi m'a^2/5,\quad K_q=64\pi D/(3a^2),\quad Q=p\pi a^2/3,\quad f_n=\sqrt{K_q/M_q}/(2\pi)$
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。
Mq = 3.1416×10⁻¹¹ kg, Kq = 804.25 N/m, fn = 0.8053 MHz, Q = 26.18 µN. Uniform loading projects with factor 1/3; concentrating the same total force at the centre approaches factor 1. These are assumed disk properties, not measured laminate coefficients.
Mq = 3.1416×10⁻¹¹ kg，Kq = 804.25 N/m，fn = 0.8053 MHz，Q = 26.18 µN。均匀载荷投影系数为 1/3；相同总力集中于中心时系数趋于 1。这些是圆盘假设参数，不是层合结构实测系数。
For Z1 = 1.5×10⁶ and Z2 = 3.0×10⁶ kg/(m² s), calculate pressure reflection/transmission and energy transmission. For layers (10 µm,2000 m/s) and (5 µm,1500 m/s), compute transit time.
Z1 = 1.5×10⁶、Z2 = 3.0×10⁶ kg/(m² s)，计算压力反射、透射及能量透射。对 (10 µm,2000 m/s) 和 (5 µm,1500 m/s) 两层，计算穿越时间。
Reference answer / 参考答案
$R_p=(Z_2-Z_1)/(Z_2+Z_1),\quad T_p=2Z_2/(Z_1+Z_2),\quad T_I=4Z_1Z_2/(Z_1+Z_2)^2,\quad t_{\mathrm{transit}}=\sum h_i/c_{L,i}$
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。
Rp = 1/3, Tp = 4/3, TI = 8/9, ttransit = 8.333 ns. Pressure amplification does not violate energy conservation. These illustrative layers differ from the source’s 29.9-ns stack; first arrival omits subsequent reflections, attenuation and cohesive response.
Rp = 1/3、Tp = 4/3、TI = 8/9、ttransit = 8.333 ns。压力幅值放大不违反能量守恒。示例层不同于来源 29.9 ns 堆叠；首波近似遗漏后续反射、衰减及黏聚响应。
Take tensile tn = 0.6 Tn0 and ts = 0.8 Ts0. Does damage initiate? For E′ = 3 GPa, Gc = 0.10 J/m² and T0 = 10 MPa estimate cohesive length. Is initiation complete separation?
拉伸 tn = 0.6 Tn0、ts = 0.8 Ts0，损伤开始吗？E′ = 3 GPa、Gc = 0.10 J/m²、T0 = 10 MPa，估算黏聚长度。起始损伤是否等于完全分离？
Reference answer / 参考答案
$(\langle t_n\rangle/T_{n0})^2+(t_s/T_{s0})^2=1,\quad \ell_{\mathrm{cz}}\sim E'G_c/T_0^2,\quad G_c=\int\mathbf t\cdot d\boldsymbol\delta$
$Structural motion and interfacial release · original governing laws; t_n: physical time; T_n0: normal cohesive strength; t_s: physical time; T_s0: shear cohesive strength; ℓ_cz: cohesive-zone length scale; E': energy / elastic modulus as defined; G_c: fracture work per release area; T_0: temperature / kinetic energy as defined; t: physical time; δ: separation / perturbation / penetration depth$
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。
0.6²+0.8² = 1, at initiation in this model. ℓcz ~ 3 µm before geometry/law coefficients. Complete separation still requires damage evolution, enough fracture work and propagation. Compressive fluid pressure cannot simply be renamed tensile cohesive traction.
0.6²+0.8² = 1，达到该模型起始条件。忽略几何与规律系数时 ℓcz ~ 3 µm。完全分离仍需损伤演化、足够断裂功及传播。不能把压缩流体压力直接改名为拉伸黏聚牵引。
For m′ = 0.020 kg/m², Gc = 0.10 J/m² and JA = 0.040 Pa s, calculate the rigid-layer ΠJ and energy-per-area ratio. Compare with the source’s independently computed ΠE.
m′ = 0.020 kg/m²、Gc = 0.10 J/m²、JA = 0.040 Pa s，计算刚性层 ΠJ 与单位面积能量比，并与来源独立计算的 ΠE 比较。
Reference answer / 参考答案
$J_{\mathrm{nom}}=\sqrt{2m'G_c},\quad \Pi_J=J_A/J_{\mathrm{nom}},\quad (E/A)/G_c=J_A^2/(2m'G_c)$
$Structural motion and interfacial release · original governing laws; J_nom: nucleation rate / areal impulse; m': mass or areal mass; G_c: fracture work per release area; Π_J: impulse availability ratio; J_A: selected areal pressure impulse; E: energy / elastic modulus as defined; A: area / volume-acceleration coefficient$
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。
Jnom = 0.06325 Pa s, ΠJ = 0.6325, ideal rigid-layer energy ratio 0.4000 = ΠJ². The source’s ΠJ = 0.565 and modal ΠE = 0.767 use different reductions, so they need not be squares. Neither is a release probability or a local energy-release rate.
Jnom = 0.06325 Pa s，ΠJ = 0.6325，理想刚性层能量比 0.4000 = ΠJ²。来源 ΠJ = 0.565 与模态 ΠE = 0.767 采用不同降阶模型，不要求平方关系。二者都不是释放概率或局部能量释放率。

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。