TENSOR de TENSIONES y vector Tension - RESISTENCIA de MATERIALES

The Stress Tensor and Stress Vector
📌 The stress tensor is a matrix containing valuable information about the stress state and distribution of stresses and strains within a continuous medium at an infinitesimal point (like a cube).
📏 Stresses are divided into normal stresses (perpendicular to the cube faces, represented by $\sigma$, e.g., $\sigma_x, \sigma_y, \sigma_z$) and tangential (shear) stresses (contained within the faces, represented by $\tau$, e.g., $\tau_{xy}, \tau_{xz}, \tau_{yz}$).
🔄 Due to equilibrium conditions (no rotation), the stress tensor is symmetric ($\tau_{ij} = \tau_{ji}$), reducing the number of independent variables from nine to six: three normal stresses and three shear stresses ($\tau_{xy}, \tau_{xz}, \tau_{yz}$).

Calculating the Stress Vector
📐 The stress vector, $\mathbf{T}$, acting on any plane passing through the point, is calculated by multiplying the stress tensor ($\boldsymbol{\sigma}$) by the unit normal vector ($\mathbf{n}$) to that plane: $\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}$.
➕ The normal vector $\mathbf{n}$ must first be normalized (made into a unit vector) by dividing its components by its magnitude.
➕ An example calculation demonstrates multiplying the stress tensor by the normalized normal vector to find the stress vector components.

Intrinsic Components of the Stress Vector
📏 The normal component of the stress vector ($\mathbf{T}_n$) is obtained by taking the dot product of the stress vector $\mathbf{T}$ with the normal vector $\mathbf{n}$: $\text{Magnitude of } T_n = \mathbf{T} \cdot \mathbf{n}$.
↪️ The tangential component of the stress vector ($\mathbf{T}_t$) can be found by subtracting the normal vector component from the total stress vector: $\mathbf{T}_t = \mathbf{T} - \mathbf{T}_n$.
🔺 Alternatively, the magnitude of the tangential component can be found using the Pythagorean theorem, as the normal and tangential vectors are perpendicular: $|\mathbf{T}|^2 = |\mathbf{T}_n|^2 + |\mathbf{T}_t|^2$.

Key Points & Insights
➡️ The stress tensor fundamentally simplifies the description of stress at a point into three normal stresses and three independent shear stresses.
➡️ The stress vector allows determination of the state of stress on *any* arbitrary plane passing through the point, provided the plane's normal direction vector is known.
➡️ Intrinsic components (normal and tangential) of the stress vector are independent of the coordinate base used for calculation.

📸 Video summarized with SummaryTube.com on Mar 01, 2026, 04:04 UTC