1.1 Introduction to Continuum Mechanics in Machine Design

In advanced machine design and forensic failure analysis, simplified one-dimensional strength-of-materials formulas are rarely sufficient. Turbine rotors, thick-walled pressure vessels, helical gear teeth under contact, crankshaft fillets, and bolted flange hubs experience genuinely three-dimensional stress and strain fields. Continuum mechanics provides the mathematical framework to describe these fields as continuous functions of position and time.

The continuum hypothesis assumes that matter is continuously distributed throughout the volume occupied by a body. Atomic discreteness is ignored so that density, displacement, stress, and strain may be differentiated. The hypothesis is valid when the characteristic length of the engineering feature (fillet radius, wall thickness, contact half-width) is much larger than grain size or mean free path. Under this abstraction, balance laws of mass, linear momentum, angular momentum, and energy close into the classical theory of elasticity (and later plasticity and fracture).

In failure analysis, continuum fields identify:

1. Local triaxiality that controls ductile vs brittle appearance.
2. Stress concentrations at notches and contacts.
3. Principal directions that align with crack paths or slip systems.
4. Hydrostatic vs deviatoric partitions that decide yielding.

1.1.1 Scope of this chapter

This chapter establishes the tensor language used throughout the course. Every subsequent chapter—yield criteria (Ch.2), fatigue (Ch.3–4), fracture (Ch.5–6), contact (Ch.7), and forensic failure analysis (Ch.14)—assumes fluency in:

• Cauchy stress and its transformation under rotation.
• Principal stresses and stress invariants.
• Hydrostatic/deviatoric decomposition.
• Infinitesimal strain and Hooke's law in 3D.

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1.2 Traction Vectors and Cauchy's Formula

Consider an imaginary cutting plane through a deformed body with unit outward normal $\hat{\mathbf{n}}$. Let $\Delta\mathbf{F}$ be the resultant force exerted on area $\Delta A$ by the material on the positive side of the plane. The traction (stress vector) is

$$\mathbf{T}^{(\hat{\mathbf{n}})} = \lim_{\Delta A \to 0} \frac{\Delta\mathbf{F}}{\Delta A} = \frac{d\mathbf{F}}{dA}.$$

Traction depends on orientation: different cutting planes through the same material point generally carry different tractions. Therefore a single vector cannot fully describe the stress state. Cauchy proved that the mapping from plane normal to traction is linear, so there exists a second-order tensor $\boldsymbol{\sigma}$ (the Cauchy stress tensor) such that

$$\mathbf{T} = \boldsymbol{\sigma}\,\hat{\mathbf{n}}.$$

In components, $T_i = \sigma_{ij} n_j$ (summation implied). This is the cornerstone relation connecting continuum stress to free-body equilibrium on arbitrary cuts.

1.2.1 Normal and shear components on an arbitrary plane

For a plane with normal $\hat{\mathbf{n}}$, the normal stress on that plane is

$$\sigma_n = \mathbf{T}\cdot\hat{\mathbf{n}} = n_i\sigma_{ij}n_j,$$

and the shear (tangential) traction magnitude follows from $\mathbf{T}_\mathrm{tan} = \mathbf{T} - \sigma_n\hat{\mathbf{n}}$. In plane stress, Mohr's circle is a graphical construction of $(\sigma_n, \tau)$ as $\theta$ varies.

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1.3 Components of the Cauchy Stress Tensor

In a Cartesian frame $(x,y,z)$ or $(x_1,x_2,x_3)$, the stress tensor has nine components arranged as

$$\boldsymbol{\sigma} = \begin{bmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix}.$$

1.3.1 Normal and shear stresses

• $\sigma_{xx},\sigma_{yy},\sigma_{zz}$ are normal stresses on the coordinate faces (positive = tension).
• $\tau_{xy},\tau_{xz},\ldots$ are shear stresses. The first index denotes the face; the second denotes the direction of the traction component.

1.3.2 Sign convention

A positive shear stress on a positive face acts in a positive coordinate direction; on a negative face it acts in a negative direction. This convention makes Mohr's circle and transformation equations consistent with continuum textbooks (e.g. Hibbeler, Shigley, Malvern).

1.3.3 Symmetry

Angular momentum balance on an infinitesimal cube (neglecting body couples) requires $\tau_{ij}=\tau_{ji}$. Hence $\boldsymbol{\sigma}$ is symmetric and only six independent components remain. Voigt notation packs them as $\{\sigma_x,\sigma_y,\sigma_z,\tau_{yz},\tau_{zx},\tau_{xy}\}$.

1.3.4 Engineering vs tensor shear

In FEA post-processors, shear stress components may be reported as tensor quantities $\tau_{ij}$ or as engineering shear strains $\gamma_{ij}=2\varepsilon_{ij}$. Always verify the convention before comparing to hand calculations or strain-gauge data.

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1.4 Stress Transformation and Mohr's Circle

When axes rotate, stress components change even though the physical state is invariant. For plane stress $(\sigma_x,\sigma_y,\tau_{xy})$, rotation by physical angle $\theta$ gives

$$\sigma_{x'} = \frac{\sigma_x+\sigma_y}{2}+\frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta,$$

$$\tau_{x'y'} = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta.$$

Define Mohr center and radius

$$C=\frac{\sigma_x+\sigma_y}{2},\qquad R=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}.$$

Then principals are $\sigma_{1,2}=C\pm R$ and $\tau_{\max}=R$ (in-plane). A physical rotation $\theta$ corresponds to a $2\theta$ rotation on Mohr's circle. Sign convention on Mohr's plane: positive shear is often plotted downward so that the circle rotation sense matches the physical rotation.

1.4.1 Principal directions

Setting $\tau_{x'y'}=0$ yields

$$\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x-\sigma_y}.$$

The two solutions $\theta_p$ and $\theta_p+90^\circ$ give the principal axes. Aligning FEA results or strain-gauge rosettes with these axes is standard practice before applying yield or fatigue criteria.

1.4.2 Three-dimensional stress transformation (brief)

For a full 3D state, rotation by direction-cosine matrix $Q_{ij}$ gives

$$\sigma'_{ij} = Q_{ik}\,\sigma_{kl}\,Q_{jl}.$$

This is a congruence transformation of a symmetric matrix. The eigenvalues (principal stresses) are invariant under all such rotations. In practice, FEA solvers compute principals directly via eigen-decomposition; hand calculations use the characteristic cubic (Section 1.5).

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1.5 Three-Dimensional Principal Stresses and Invariants

In 3D, principals are eigenvalues of $\boldsymbol{\sigma}$:

$$\det(\boldsymbol{\sigma}-\sigma\mathbf{I})=0 \quad\Rightarrow\quad \sigma^3 - I_1\sigma^2 + I_2\sigma - I_3 = 0,$$

with invariants

$$I_1=\mathrm{tr}\,\boldsymbol{\sigma}=\sigma_{xx}+\sigma_{yy}+\sigma_{zz},$$

$$I_2=\sigma_{xx}\sigma_{yy}+\sigma_{yy}\sigma_{zz}+\sigma_{zz}\sigma_{xx}-\tau_{xy}^2-\tau_{yz}^2-\tau_{zx}^2,$$

$$I_3=\det\boldsymbol{\sigma}.$$

Ordered principals satisfy $\sigma_1\ge\sigma_2\ge\sigma_3$. The maximum shear stress in 3D is $\tau_{\max}=(\sigma_1-\sigma_3)/2$. These quantities are coordinate-free and feed directly into Tresca, von Mises, and fracture criteria.

1.5.1 Octahedral (mean normal and shear) stresses

On planes equally inclined to all principal axes (octahedral planes), the normal and shear stresses are

$$\sigma_{\mathrm{oct}} = \frac{1}{3}(\sigma_1+\sigma_2+\sigma_3) = \sigma_m,$$

$$\tau_{\mathrm{oct}} = \frac{1}{3}\sqrt{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2} = \sqrt{\frac{2}{3}J_2}.$$

Note that $\bar{\sigma}_{\mathrm{vm}} = \sqrt{3}\,\tau_{\mathrm{oct}}$. Octahedral quantities appear in crystal plasticity and in some multiaxial fatigue models.

1.5.2 Stress triaxiality

The ratio $\eta = \sigma_m / \bar{\sigma}_{\mathrm{vm}}$ (or variants using $\sigma_1$) quantifies how "hydrostatic" a stress state is. High triaxiality under tension promotes void nucleation and ductile dimple fracture; low triaxiality favors shear-dominated slip. This connects directly to Ch.6 EPFM and Ch.14 forensic interpretation.

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1.6 Hydrostatic and Deviatoric Decomposition

Any stress tensor splits uniquely as

$$\boldsymbol{\sigma}=\sigma_m\mathbf{I}+\mathbf{s},\qquad \sigma_m=\frac{1}{3}I_1,\qquad \mathbf{s}=\boldsymbol{\sigma}-\sigma_m\mathbf{I}.$$

• Hydrostatic part $\sigma_m\mathbf{I}$ changes volume (dilatation) but not shape.
• Deviatoric part $\mathbf{s}$ changes shape and drives dislocation plasticity in metals.

The second invariant of $\mathbf{s}$ is

$$J_2=\frac{1}{2}s_{ij}s_{ij}=\frac{1}{6}\big[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\big].$$

Von Mises stress is $\bar{\sigma}_{\mathrm{vm}}=\sqrt{3J_2}$. This decomposition is mandatory before Ch.2 yield criteria.

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1.7 Infinitesimal Strain Tensor

For small deformations, the displacement gradient splits into strain and rotation:

$$\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}),\qquad \omega_{ij}=\frac{1}{2}(u_{i,j}-u_{j,i}).$$

Engineering shear strain is $\gamma_{xy}=2\varepsilon_{xy}$. The strain tensor is symmetric like stress; principal strains and strain invariants are defined analogously.

1.7.1 Compatibility equations (Saint-Venant)

Six strain components cannot be arbitrary—they must derive from a single-valued displacement field. The compatibility equations (simplified form):

$$\frac{\partial^2\varepsilon_{xx}}{\partial y^2}+\frac{\partial^2\varepsilon_{yy}}{\partial x^2}=\frac{\partial^2\gamma_{xy}}{\partial x\,\partial y},$$

plus five similar relations. In FEA, compatibility is enforced automatically by shape functions that interpolate displacement. In experimental mechanics, inconsistent strain fields (e.g. from noisy DIC) may violate compatibility—a sign of measurement error.

1.7.2 Volumetric and deviatoric strain

Analogous to stress,

$$\varepsilon_v = \varepsilon_{kk} = \varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz},\qquad e_{ij} = \varepsilon_{ij} - \frac{1}{3}\varepsilon_v\delta_{ij}.$$

Volumetric strain drives pressure change; deviatoric strain drives shape change. In plasticity, volumetric plastic strain is often negligible for metals (incompressible plastic flow).

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1.8 Linear Isotropic Hooke's Law

For isotropic linear elasticity,

$$\varepsilon_{ij}=\frac{1+\nu}{E}\sigma_{ij}-\frac{\nu}{E}\sigma_{kk}\delta_{ij},$$

or inversely

$$\sigma_{ij}=\lambda\varepsilon_{kk}\delta_{ij}+2G\varepsilon_{ij},$$

with Lamé parameters $\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}$ and $G=\frac{E}{2(1+\nu)}$. Related moduli:

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1.9 Plane Stress vs Plane Strain

These two idealizations reduce 3D problems to 2D and appear constantly in machine design.

1.9.1 Plane stress

Assumption: $\sigma_z = \tau_{xz} = \tau_{yz} = 0$ (thin plate, loaded in its plane).

Hooke's law in plane stress:

$$\varepsilon_x = \frac{1}{E}(\sigma_x - \nu\sigma_y),\quad \varepsilon_y = \frac{1}{E}(\sigma_y - \nu\sigma_x),\quad \gamma_{xy} = \frac{\tau_{xy}}{G}.$$

The out-of-plane strain $\varepsilon_z = -\frac{\nu}{E}(\sigma_x+\sigma_y) \neq 0$ even though $\sigma_z=0$. Thin webs, gear faces, and plate elements in FEA use plane stress.

1.9.2 Plane strain

Assumption: $\varepsilon_z = \gamma_{xz} = \gamma_{yz} = 0$ (long body, constrained normal strain).

Then $\sigma_z = \nu(\sigma_x+\sigma_y)$ (not zero). Thick cylinders, rolling contact strips, and long extrusions approximate plane strain. Plane strain is stiffer than plane stress for the same in-plane loads.

1.9.3 When to use which

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1.10 Strain Gauges and Rosettes

Resistance strain gauges measure average strain over a small gauge length (typically 1–6 mm). A single gauge gives one normal strain component; a rosette (2- or 3-gauge array) recovers principal strains and directions.

1.10.1 Single gauge

If a gauge is aligned at angle $\alpha$ from the $x$-axis, it measures

$$\varepsilon_\alpha = \varepsilon_x\cos^2\alpha + \varepsilon_y\sin^2\alpha + \frac{\gamma_{xy}}{2}\sin 2\alpha.$$

1.10.2 Three-gauge rectangular rosette (0°, 45°, 90°)

From measured $\varepsilon_a, \varepsilon_b, \varepsilon_c$:

$$\varepsilon_x = \varepsilon_a,\quad \varepsilon_y = \varepsilon_c,\quad \gamma_{xy} = 2\varepsilon_b - \varepsilon_a - \varepsilon_c.$$

Principal strains follow Mohr's circle for strain (with $\gamma/2$ as the shear ordinate). Stresses are recovered via Hooke's law only if the material is linear elastic and the stress state is known to be plane stress.

1.10.3 Delta rosette (0°, 60°, 120°)

Preferred when principal direction is unknown. The three equations in three unknowns $(\varepsilon_x, \varepsilon_y, \gamma_{xy})$ are solved simultaneously. Rosette data is essential for validating FEA at critical locations (fillet roots, bolt holes).

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1.11 FEA Post-Processing Notes

Commercial FEA packages (ANSYS, Abaqus, Nastran) report stress and strain in several coordinate systems. Misinterpretation is a common source of design errors.

1.11.1 Coordinate systems

• Global: fixed lab frame; useful for assembly-level checks.
• Local (element): varies per element; needed for orthotropic materials.
• Principal: eigenvectors of $\boldsymbol{\sigma}$; use for yield and fatigue.

1.11.2 Stress types in FEA

• Nodal-averaged: smoothed across elements; good for trends, can hide singularities.
• Element (centroid): discontinuous across elements; more conservative at hotspots.
• Extrapolated from Gauss points: default in many codes; check documentation.

1.11.3 Singularities and mesh convergence

At sharp re-entrant corners, $\sigma \to \infty$ as mesh refines (elastic solution). Do not compare raw peak stress to yield without understanding singularity or using stress concentration factors from handbook data. For fillets, refine until principal stress at a fixed distance from the root converges.

1.11.4 Recommended post-processing workflow

1. Identify hotspot (von Mises or max principal contour).
2. Extract principal stresses $\sigma_1, \sigma_2, \sigma_3$ at the node/element of interest.
3. Compute hydrostatic and deviatoric parts.
4. Apply appropriate failure theory (Ch.2–6).
5. Compare with strain-gauge or photoelastic data if available.

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1.12 Engineering Applications and Failure Context

• Shafts under bending + torsion: Mohr circle from $\sigma=Mc/I$ and $\tau=Tr/J$ → principals for fatigue (Ch.3).
• Pressure vessels: $\sigma_h,\sigma_l,\sigma_r$ as principals; triaxiality affects toughness requirements (Ch.5).
• Contact (Ch.7): subsurface principals rotate with depth; $\tau_{\max}$ location predicts spalling.
• Bolted joints (Ch.12): combined tension + bending + shear at thread root; 3D FEA essential.
• Forensics (Ch.14): fracture surface orientation vs principal directions confirms Mode I vs mixed mode; beach marks align perpendicular to $\sigma_1$ in fatigue.

1.12.1 Ductile vs brittle failure signatures

1.12.2 Design takeaway

Always reduce a 3D FEA or analytical field to principals + hydrostatic/deviatoric split before applying yield, fatigue, or fracture theories. Never apply a uniaxial allowable directly to a multi-axial hotspot without transformation.