Lecture 10: Masses, Forces, and Stresses

Overview

Steven M. Wise introduces continuum mechanics concepts for mass and force, defining mass density, body forces, surface tractions, and their resultant forces and torques. Using Cauchy's postulate and theorem, he represents traction as t(n,x) = S(x)n, then derives the local equilibrium conditions div S + ρb = 0 and symmetric stress tensor S from force and torque balance.

Key takeaways

For a smooth positive density ρ, the mass of a region Ω is ∫Ω ρ dV, and the center of mass agrees with the center of volume only when ρ is uniform.
Body-force density is measured in newtons per cubic meter; dividing it by mass density gives specific force in newtons per kilogram, with gravity as a standard example.
Cauchy's postulate makes traction depend only on the local point and plane normal, and reversing the normal gives equal-and-opposite tractions across an internal surface.
Cauchy's theorem compresses the traction field into the stress tensor relation t(n,x) = S(x)n, with each tensor column representing traction on a coordinate plane.
Force balance for every subregion localizes to div S + ρb = 0, converting an integral equilibrium condition into a differential equation.
Torque balance, together with force balance, requires the Cauchy stress tensor to be symmetric.

Chapters

0:00 Regular Regions, Boundaries, and Orientability

A regular region is a finite union of disjoint, open, bounded components with piecewise-smooth boundaries.
The definition permits holes, so boundaries can have multiple disjoint components, such as an outer surface and an inner surface.
Each boundary component must be orientable, giving it distinct sides and allowing a consistent unit normal.

2:27 Mass Density and the Mass Integral

A mass density ρ is a smooth, positive scalar field defined on a body, with SI units of kilograms per cubic meter.
The mass of any subset Ω is m(Ω) = ∫Ω ρ dV, while its volume is ∫Ω 1 dV.
Mass density weights each volume element's contribution to the total mass.

4:34 Continuum Assumption and Pointwise Density

Pointwise density can be understood as the limit of mass divided by volume for shrinking neighborhoods around a point.
A ball of radius δ illustrates the shrinking-volume construction as δ approaches zero.
The continuum model treats material as present at every point, even though real solids consist of atoms and the approximation eventually breaks down at small scales.

7:55 Center of Mass and Center of Volume

The center of mass is x̄ = (1/m(Ω))∫Ω xρ(x) dV, a density-weighted first moment.
The center of volume is (1/Vol(Ω))∫Ω x dV; it coincides with the center of mass when ρ is uniform.
For disconnected regions, such as two separated spheres of equal uniform density, either center can lie outside the material.

10:31 Body Forces and Specific Force

A body-force field is a vector field measured in newtons per cubic meter, and integrating it over Ω gives a resultant force in newtons.
The resultant torque about a point z is ∫Ω (x − z) × b̂ dV, with units of newton-meters.
Specific body force b is force per unit mass, measured in newtons per kilogram; the force per volume is ρb.
Gravity is the principal example of a body force acting throughout a material rather than through surface contact.

17:15 Surface Tractions Across Internal and External Boundaries

An internal surface γ divides a body into B− and B+; its chosen unit normal points from B− toward B+.
The traction tₙ is the force per unit area exerted by the positive side on the negative side, and it may vary in direction across γ.
On an external boundary, the convention is to take the unit normal outward from the body.
Traction has the same units as pressure: pascals, or newtons per square meter.

24:36 Resultant Force and Torque from Traction

The resultant surface force on γ is ∫γ tₙ dA, converting force per area into newtons.
The resultant torque about z is ∫γ (x − z) × tₙ dA, measured in newton-meters.
These surface integrals complement volume integrals of body forces when calculating a body's total force and torque.

26:48 Cauchy's Postulate and Equal-and-Opposite Tractions

Cauchy's postulate says traction at point x depends only on the local unit normal n and x, through a traction function t(n,x).
Under the stated continuity and non-concentration assumptions, reversing the normal reverses the traction: t(−n,x) = −t(n,x).
The result expresses action and reaction across an internal cut: each side exerts an equal and opposite force on the other.
The non-concentration assumption rules out force distributions concentrated at a single point like a Dirac delta.

37:08 Cauchy's Theorem Defines the Stress Tensor

Cauchy's theorem states that traction is linear in the surface normal, so a second-order stress tensor S(x) satisfies t(n,x) = S(x)n.
The tensor maps each possible plane normal to the traction acting across that plane at the same point.
In an orthonormal frame, traction components obey tᵢ = Sᵢⱼ nⱼ, using the repeated-index summation convention.
The theorem's proof is assigned to the text, while its tensor interpretation is developed in the lecture.

40:06 Stress Tensor Columns and Plane-Dependent Traction

At a fixed point x, changing the dividing plane changes its normal and generally changes the traction.
Setting n = eⱼ shows that the jth column of S contains the traction components on the coordinate plane normal to eⱼ.
Stress components vary with position, because the traction function and stress tensor both depend on x.
The tensor representation lets one recover traction on any oriented plane from the same local stress state.

49:49 Equilibrium Examples: Clamped Solids and Jello

A clamped elastic body pulled by a top-boundary traction raises questions about its displacement, stress, and strain at equilibrium.
A piece of jello resting on a table deforms under gravity, illustrating equilibrium driven by a body force rather than an applied traction.
The class aims to derive equations that predict equilibrium from density, applied tractions, and specific body forces.

53:02 Integral Force and Torque Balance for Every Subregion

Mechanical equilibrium requires zero resultant total force and zero resultant torque for every open subset Ω of the body.
Total force combines the volume term ∫Ω ρb dV with the boundary traction term ∫∂Ω t dA.
Torque balance adds the moments of both body forces and boundary tractions about a point z.
If resultant force is zero, torque is independent of the reference point, so it is sufficient to test torque about the origin.

58:24 Local Force Balance: Divergence of Stress Plus Body Force

The global force balance becomes local when traction is replaced by Sn and the tensor divergence theorem converts the boundary integral into a volume integral.
Because the balance holds for every Ω, the localization theorem implies div S + ρb = 0 at each point.
This Cauchy equilibrium equation is a differential condition requiring smooth stress, density, and specific body-force fields.
Wise derives the force equation from integral balance and notes that the reverse implication is the easier direction left as an exercise.

1:04:09 Torque Balance Forces the Stress Tensor to Be Symmetric

Torque balance is rewritten using t = Sn, then the tensor divergence theorem and localization reduce it to a local identity.
Wise introduces a tensor R using the permutation symbol εᵢⱼₖ to express the moment term x × Sn and apply the divergence theorem.
After applying the product rule, the force-balance terms cancel, leaving εᵢⱼₖ Sₖⱼ = 0.
Evaluating the free index for the three coordinate directions gives S₃₂ = S₂₃, S₁₃ = S₃₁, and S₂₁ = S₁₂, so S is symmetric.

Transcript excerpt

0:04 Welcome back to continue mechanics. Then we'll get started with chapter 3. Today we're going to be talking about or in uh chapter 3 we'll be talking about continuum concepts of math and force. I think we'll probably spend two to three lectures in here if I had to guess. Probably three. So this is part one of three. The first thing we want to talk about is mass density. But before I get there, let's review uh what we mean by a regular region because we're going to be using this again and again. So a region is called regular if and only if it is the union of a finite number of open, disjoint and bounded components.

0:54 The boundary surface which is always uh labeled partial B is pie-wise smooth and consists of a finite number of disjoint components. This assumption here allows the fact that we could have holes in our in our material because think about it even in 2D if we have a an object like this then it has a pie-wise smooth boundary and it has uh disjoint connected components for the boundary. So here's one kind of like the outer boundary and then there's an inner boundary of this. So it can have holes. Uh three, each component of the boundary

1:45 is orientable in the sense that it has two distinct sides. So this is more important for uh three dimensions. So you don't want to have a surface describing the uh outer boundary of this object which is something akin to the mobius strip where you know you could start pointing out on up on one side and then come around and you know go make a loop around or a transit around it and come up on the other side pointing in

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.