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Slip-line field theory solves certain two-dimensional problems of plastic deformation by constructing two orthogonal families of characteristic curves, called slip lines. Its classical assumptions include plane strain, negligible inertia and body force, and a rigid, perfectly plastic material. The curves are aligned with the directions of maximum in-plane shear stress, rather than the principal stress directions.^[ref]

The curve network encodes both the stress field and constraints on admissible velocity. Indentation of a thick workpiece by a rigid flat punch provides a motivating application: the construction predicts a plastic flow region and the load needed to sustain that flow. It is useful because geometry can turn coupled field equations into relations integrated along curves. A slip line is a continuum characteristic, however, not necessarily a microscopic crystal slip system or the actual path of a moving material particle.

A solid resists sustained changes of shape on the timescale being considered. It can support a static shear stress, whereas an ordinary liquid continues to deform under sustained shear. Small loads often produce recoverable elastic deformation; larger loads can cause plastic deformation or fracture.

A solid need not be a crystal. A crystalline specimen has long-range structural order, while an amorphous solid such as ordinary glass lacks the same repeating lattice arrangement. The mechanical classification and the structural classification answer different questions. Applying a small lateral load to the top of a block while holding its base fixed tests shear resistance; a finite shear displacement at static balance is a solid-like response. At long times some solids also creep, so the observation interval and applied stress matter when comparing a material's response with an ideal elastic model.

The tangent modulus is the local slope of a specified uniaxial stress-strain curve:It converts a small additional strain near ${\epsilon }_0$ into the corresponding first-order change of stress. For nonlinear elastic deformation, this slope can vary with the operating point.^[ref] In uniaxial linear elasticity, $\sigma =E\epsilon$, so the tangent modulus equals the constant Young's modulus.^[ref]

For an original nonlinear branch, stipulateThen $E_{\tan }=E_0+2B\epsilon$. At ${\epsilon }_0=0.02$, the stress is $80\mathrm{MPa}$ and the tangent modulus is $6\mathrm{GPa}$. The secant modulus from the stipulated unloaded origin is instead $\sigma /{\epsilon }_0=4\mathrm{GPa}$. The two slopes answer different questions: the secant compares total stress and strain, while the tangent predicts a small further increment.

For $\Delta \epsilon =0.001$, the tangent predicts $\Delta \sigma \simeq 6\mathrm{MPa}$. The exact change in the chosen polynomial is $6.1\mathrm{MPa}$ because the omitted term is $B(\Delta \epsilon {)}^2=0.1\mathrm{MPa}$. This is a linear approximation whose error decreases quadratically with the increment, not a claim that the nonlinear material obeys Hooke's law globally.

To estimate a tangent modulus experimentally, specify the stress and strain measures, temperature, loading history and loading rate. For engineering stress $\sigma =F/A_0$ and engineering strain $\epsilon =\Delta L_g/L_g$ from a local extensometer,Fit a local window about the chosen operating point and check how the estimate changes with that window. For an original $L_g=50\mathrm{mm}$, $A_0=10\mathrm{m}{\mathrm{m}}^2$ and local slope $1200\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$, this gives $6000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-2}=6\mathrm{GPa}$. Differentiation amplifies measurement noise, and crosshead displacement can include machine compliance. Where plastic deformation or history-dependent response occurs, identify the loading branch: its local slope need not equal the unloading slope or an elastic material constant.

Tomasz Wierzbicki is an applied-mechanics researcher and MIT emeritus professor whose work concerns plastic deformation, structural failure and crashworthiness. His MIT curriculum vitae lists research comparing experimental sheet fracture and tearing with numerical predictions; the laboratory account identifies dynamic plasticity and structural failure as central research themes.^[ref]

His structural-mechanics teaching makes the relation between measurement and model explicit. A uniaxial tensile test measures axial force, extension and lateral contraction. Those observations determine Young's modulus and Poisson's ratio under the stated loading conditions, rather than treating material parameters as properties of a specimen's shape.^[ref]

An original work calculation illustrates a related structural-design distinction. A constant resisting force of $3\mathrm{kN}$ over a compression stroke of $0.1\mathrm{m}$ takes input work $300\mathrm{J}$. A triangular force–displacement curve with the same peak and stroke has area $150\mathrm{J}$. Equal peak loads therefore need not imply equal energy input. The area under the measured curve, together with unloading behaviour, helps distinguish total work from recoverable elastic energy.

These are illustrative curves, not measurements from a Wierzbicki experiment. His work on structural failure and constitutive models addresses the broader task of connecting geometry, material response and observed load histories.

Work hardening is an increase in resistance to further plastic deformation caused by earlier deformation. A useful macroscopic test loads a metal beyond yield, unloads it and reloads in the same direction: further plastic flow then requires a larger stress. In a constitutive equation, this history is represented by evolving internal variables rather than by the current elastic strain alone.^[ref]

In a crystal, multiplication and interaction of dislocations can make later dislocation glide more difficult. Forest hardening and latent hardening describe particular aspects of that obstacle formation. Increased resistance to plastic flow is not the same as a large change in Young's modulus: the elastic unloading slope and the load required to restart permanent deformation measure different properties. Work hardening can delay necking by compensating for the shrinking loaded area.