Biomechanical Modeling

A set of three physical laws forming the basis of classical mechanics. The first law (inertia) states an object remains at rest or in uniform motion unless acted upon by a net external force. The second law quantifies force as mass times acceleration, [latex]\vec{F} = m\vec{a}[/latex]. The third law states that for every action, there is an equal and opposite reaction.

A Newtonian fluid's shear stress is directly proportional to the rate of shear strain. This linear relationship is defined by Newton's law of viscosity: [latex]\tau = \mu \frac{du}{dy}[/latex], where [latex]\tau[/latex] is the shear stress, [latex]\mu[/latex] is the dynamic viscosity (a constant of proportionality), and [latex]\frac{du}{dy}[/latex] is the shear rate or velocity gradient.

Bernoulli's principle states that for an inviscid flow, an increase in a fluid's speed occurs simultaneously with a decrease in pressure or a decrease in its potential energy. It is a statement of the conservation of energy for a moving fluid, commonly expressed as [latex]p + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}[/latex] along a streamline.

流体力学是应用力学的一个分支，研究液体和气体的静力学（静止流体）和动力学（运动流体）。它运用质量、动量和能量守恒的基本原理来分析和预测流体行为。其应用范围广泛，涵盖空气动力学、水力学、气象学和海洋学等诸多领域。

In continuum mechanics, the principle of mass conservation states that the mass of a closed system must remain constant over time. For a fluid, this is expressed by the continuity equation. In its Eulerian differential form, it is written as [latex]\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0[/latex], where [latex]\rho[/latex] is the density and [latex]\mathbf{u}[/latex] is the velocity field.

这是描述连续力学中运动的两种方法：

• 拉格朗日规范遵循单个材料粒子，跟踪它们随时间的变化，就像观察交通中的特定汽车一样
• 欧拉规范关注空间中的固定点，观察通过这些点的任何粒子的属性（速度、密度），就像交通摄像头观察固定交叉路口一样。

固体力学是连续介质力学的一个分支，研究固体材料的行为，特别是它们在力、温度变化或其他外部载荷作用下的运动和变形。它是工程结构设计和分析的基础。关键领域包括弹性力学（可恢复变形）、塑性力学（永久变形）和断裂力学（裂纹的萌生和扩展）。

Generalized Hooke's Law is the constitutive equation for linear elastic materials, stating that the stress tensor is linearly proportional to the strain tensor. The relationship is expressed as [latex]\sigma = C : \varepsilon[/latex], where [latex]\sigma[/latex] is the stress tensor, [latex]\varepsilon[/latex] is the strain tensor, and [latex]C[/latex] is the fourth-order stiffness tensor containing the material's elastic constants.

This law states that every particle attracts every other particle in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is [latex]F = G \frac{m_1 m_2}{r^2}[/latex], where [latex]G[/latex] is the gravitational constant. It unified terrestrial and celestial mechanics.

In an isolated system, the total momentum remains constant. An isolated system is one not subject to external forces. This principle follows from Newton's laws of motion. For a system of particles, the total momentum [latex]\vec{p}_{\text{total}} = \sum_{i} m_i \vec{v}_i[/latex] is conserved if the net external force is zero. This is fundamental to analyzing collisions.

In a reference frame rotating with an angular velocity [latex]\boldsymbol{\omega}[/latex], the centrifugal force [latex]\mathbf{F}_{\mathrm{cf}}[/latex] acting on an object of mass [latex]m[/latex] at a position vector [latex]\mathbf{r}[/latex] from the origin is given by the vector formula: [latex]\mathbf{F}_{\mathrm{cf}} = -m \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})[/latex]. This formula shows the force is directed perpendicular to the axis of rotation and outwards.

Coulomb's Law quantifies the electrostatic force between two stationary, electrically charged particles. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is [latex]F = k_e \frac{q_1 q_2}{r^2}[/latex]. This force can be either attractive or repulsive.

A reformulation of classical mechanics based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy ([latex]L = T - V[/latex]). The equations of motion are derived from the Euler-Lagrange equation, [latex]\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0[/latex], using generalized coordinates, which simplifies analysis of complex systems with constraints.

Galvanic corrosion is an electrochemical process where one metal corrodes preferentially when in electrical contact with another in the presence of an electrolyte. This occurs because the dissimilar metals create a bimetallic couple, with the less noble (more active) metal acting as the anode and corroding, while the more noble metal acts as the cathode.