Strain Transformation
Transforming strain at a point, strain-gage rosettes, generalized Hooke's law, and the theories of failure.
Chapter Objectives
- Distinguish a surface plane-stress measurement from true plane strain and explain the Poisson response.
- Transform normal and engineering shear strain to a specified orientation.
- Calculate principal strains, maximum shear strain, and their directions with correct engineering-shear factors.
- Reduce rosette readings to a local strain state and select compatible elastic constitutive relations.
- Use principal stress information as one input to an appropriate, limited failure screening.
10.1Plane Strain
Plane Strain
The general strain state at a point has three normal strains (volume change) and three shear strains (shape change). Plane strain keeps only \(\epsilon_x, \epsilon_y, \gamma_{xy}\). Important subtlety: plane stress does not imply plane strain — by the Poisson effect, in-plane stresses \(\sigma_x, \sigma_y\) also produce an out-of-plane strain \(\epsilon_z \ne 0\). True plane strain is rare in practice; the analysis matters mainly because it converts strain-gage data measured on a body's surface into the plane stress at that point.
10.2General Equations of Plane-Strain Transformation
Sign Convention
Normal strains positive if they cause elongation along their axes; shear strain \(\gamma_{xy}\) positive if the interior right angle between the +x and +y sides closes; θ positive counterclockwise. These match the plane-stress conventions: positive stresses cause positive strains.
Tracking the elongation and rotation of line segments dx′ and dy′ as the element deforms (each receives contributions from \(\epsilon_x\), \(\epsilon_y\), and \(\gamma_{xy}\)) yields equations identical in form to the stress transformation — with σ → ε and τ → γ/2:
- \(\epsilon_{y'}\) follows by substituting θ + 90°; the invariant \(\epsilon_{x'} + \epsilon_{y'} = \epsilon_x + \epsilon_y\) holds.
- All plane-stress machinery transfers by the substitutions \(\sigma \to \epsilon\), \(\tau \to \gamma/2\).
- No shear strain accompanies the principal strains; the max-shear element sits 45° away and also carries \(\epsilon_{avg}\).
Problem-solving strategy
- Decide whether readings describe a surface plane-stress state, true plane strain, or a general three-dimensional condition.
- Draw axes and the positive angle-closing convention; label engineering shear \(\gamma_{xy}\), not tensor shear.
- Transform directly for a named direction, or use the strain-circle radius for principal values and maximum shear strain.
- For a rosette, first solve its three independent measurement equations, then transform or use Hooke's law if material data and boundary conditions justify it.
- Check dimensionless strain units, the normal-strain sum, orientation, and elastic-range admissibility.
10.3Mohr's Circle — Plane Strain
The Strain Circle
Because the strain transformation equations have the same mathematical form as the stress equations, eliminating θ again produces a circle — plotted with \(\epsilon\) horizontal (positive right) and \(\gamma/2\) vertical (positive down):
\[ \left(\epsilon_{x'} - \epsilon_{avg}\right)^2 + \left(\frac{\gamma_{x'y'}}{2}\right)^2 = R^2, \qquad R = \sqrt{\left(\frac{\epsilon_x - \epsilon_y}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2} \]Center at \(C(\epsilon_{avg}, 0)\), reference point \(P(\epsilon_x, \gamma_{xy}/2)\). Everything reads exactly as for stress: σ-axis intersections = principal strains, extreme vertical points = \(\gamma_{max}/2\), and element rotations of θ map to circle rotations of 2θ in the same sense.
10.4Absolute Maximum Shear Strain
Out-of-Plane Strain Circles
Just as with stress, three Mohr's circles exist — one for each pair of principal strains — and the absolute maximum shear strain is the diameter of the largest circle. For plane strain (\(\epsilon_z = 0\)):
\[ \epsilon_1, \epsilon_2\ \text{same sign:}\quad \gamma_{\substack{abs\\max}} = \epsilon_1 \qquad\qquad \epsilon_1, \epsilon_2\ \text{opposite signs:}\quad \gamma_{\substack{abs\\max}} = \epsilon_1 - \epsilon_2 \]When the in-plane principal strains share a sign, the governing circle pairs \(\epsilon_1\) with the zero out-of-plane strain, and the critical shear strain acts out of plane. Note that for plane stress, \(\epsilon_z \ne 0\) (Poisson), so the third principal strain must be computed before comparing circles.
10.5Strain Rosettes
Why Rosettes?
A strain gage measures normal strain in one direction only — shear strain cannot be measured directly. A strain rosette is a cluster of three gages at angles \(\theta_a, \theta_b, \theta_c\); writing the transformation equation \(\epsilon = \epsilon_x\cos^2\theta + \epsilon_y\sin^2\theta + \gamma_{xy}\sin\theta\cos\theta\) for each reading gives three equations for the three unknowns \(\epsilon_x, \epsilon_y, \gamma_{xy}\).
10.6Material Property Relationships
For homogeneous, isotropic, linear elastic material, the strain produced by triaxial stress follows by superposing each stress's direct effect (\(\sigma/E\)) and Poisson contractions (\(-\nu\sigma/E\)):
- Shear stresses cause only shape change (no volume change), so the normal and shear relations are uncoupled.
- Setting pure shear principal stresses (\(\pm\tau_{xy}\)) into these equations is one way to derive \(G = E/[2(1 + \nu)]\) — the three constants are not independent.
- \(e\) = volumetric strain (change in volume per unit volume), valid for small strains.
- \(k\) = bulk modulus: the ratio of hydrostatic pressure to volume decrease (\(\sigma_1 = \sigma_2 = \sigma_3 = -p\)). As \(\nu \to 0.5\), \(k \to \infty\): the material becomes incompressible (rubber-like) — hence the physical bound \(\nu \le 0.5\).
10.7Theories of Failure
Failure under uniaxial stress is easy to define — yielding for ductile materials, fracture for brittle ones. Under multiaxial stress, a failure theory compares the principal stresses at the critical point against the simple-test strength. No single theory fits every material in every condition (temperature, load rate, environment all change behavior).
Ductile yielding is caused by crystal slip along planes of maximum shear (visible as Lüders lines at 45° in tension tests). Yielding begins when the absolute maximum shear stress reaches \(\sigma_Y/2\), the value at yield in simple tension. For plane stress:
\[ |\sigma_1| = \sigma_Y \;\;\text{(same signs)} \qquad\qquad |\sigma_1 - \sigma_2| = \sigma_Y \;\;\text{(opposite signs)} \]Plotted in the σ1–σ2 plane this is a hexagon; stress points inside are safe.
Materials do not yield under hydrostatic stress, so only the distortion (shape-changing) part of the strain energy drives yielding. Yielding occurs when the distortion energy per unit volume reaches its value at yield in simple tension. For plane stress:
\[ \sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2 = \sigma_Y^2 \]— an ellipse circumscribing the Tresca hexagon. The two theories agree at uniaxial and equibiaxial points and differ most in pure shear, where von Mises predicts yield at \(\tau = \sigma_Y/\sqrt{3}\), about 15 % above Tresca's \(\sigma_Y/2\) — and torsion tests show von Mises is the more accurate. Tresca is the simpler, conservative choice.
Maximum normal stress theory (Rankine): brittle fracture (sudden, tension-driven, with little yielding) occurs when the largest principal stress reaches the ultimate stress from a tension test: \(|\sigma_1| = \sigma_{ult}\) — a square boundary in the σ1–σ2 plane. Suited to materials with equal tension/compression behavior.
Mohr's failure criterion: for brittle materials with different tensile and compressive strengths (e.g. gray cast iron, concrete), failure envelopes built from tension, compression, and torsion test circles set the limit. Caveat: brittle fracture starts at microscopic flaws, so these theories scatter — use generous factors of safety.
Recognition strategy
A single gage direction requests transformed normal strain; it does not require a stress calculation or a principal-strain search.
Given information
- Strains are dimensionless and reported in microstrain; positive engineering shear closes the positive right angle.
Model and interpretation
Use small-strain kinematics on a surface element and take \(\theta=+30^\circ\) from \(x\) to the gage axis.
Verification
Units: Strain and microstrain are dimensionless ratios; the sine and cosine are dimensionless, so every added term consistently remains \(10^{-6}\).
Signs and direction: The positive result follows from positive direct strains and the positive shear contribution at \(30^\circ\); reversing the gage direction by \(180^\circ\) would not change it.
Plausibility: \(411.6\times10^{-6}\) is near the larger axial strain and below the principal maximum, which is consistent with a direction close to \(x\).
Engineering interpretation
This is the extension a correctly aligned uniaxial gage would sense. It is not a stress until elastic constants and the surface plane-stress boundary condition are established.
Recognition strategy
Three noncollinear normal readings are the signature of rosette reduction, followed by strain transformation for principal values.
Given information
- The gages are accurately aligned at \(0^\circ,45^\circ,90^\circ\); the panel is in its small-strain elastic operating range.
Model and interpretation
Take the first and third gages as \(x\) and \(y\). Engineering shear is recovered from the \(45^\circ\) compatibility equation.
Verification
Units: All rosette equations combine microstrain, a dimensionless unit; the radius and both principal strains consequently remain microstrain.
Signs and direction: Positive recovered shear means the chosen \(x\)-\(y\) angle closes under the stated convention; the principal direction is \(\theta_{p1}=57.0^\circ\), where shear vanishes.
Plausibility: The principal values bracket both direct strains and sum to \(440\ \mu\varepsilon\), exactly matching \(180+260\ \mu\varepsilon\).
Engineering interpretation
The gage data show meaningful shear as well as extension. Before using these values for an allowables decision, confirm alignment, thermal compensation, and that the gage footprint does not span a steep strain gradient.
Recognition strategy
Known principal surface strains plus elastic constants and a free surface call for inverse plane-stress Hooke law, then a ductile yield comparison.
Given information
- The surface normal stress is zero; material is homogeneous, isotropic, linear elastic, and the stated yield strength is for the relevant condition.
Model and interpretation
Principal strain axes are also principal stress axes under isotropic elasticity. Use plane stress, not plane strain, so \(\sigma_3=0\).
Verification
Units: GPa multiplied by dimensionless strain produces stress; converting \(200\ \text{GPa}\) to \(200000\ \text{MPa}\) yields MPa throughout the yield calculation.
Signs and direction: Although \(\epsilon_2\) is negative, \(\sigma_2\) is tensile because the larger \(\epsilon_1\) produces Poisson coupling; this is consistent with plane stress.
Plausibility: The equivalent stress is below \(\sigma_1\) for two same-sign principal stresses, and the factor \(2.13\) is greater than one without claiming fatigue or code compliance.
Engineering interpretation
This preliminary result indicates no predicted first yield under the supplied static load. It cannot approve a component without stress concentration, residual stress, fatigue spectrum, temperature, uncertainty, and governing-code review.
Chapter recap
- Transform strain with \(\gamma/2\), and keep strain distinct from displacement and stress.
- Reduce rosettes before selecting principal directions or a constitutive model.
- Use plane-stress boundary conditions for a free surface and calculate out-of-plane strain when needed.
- Before moving on: identify the measurement model, compute and audit a strain state, and state the limits of any failure screen.