Introduction to Deformable Bodies and Stress-Strain Relationships

Vocabulary flashcards covering solid mechanics fundamentals including rigid vs. deformable bodies, stress, strain, constitutive laws, material properties, and stress-strain curve analysis.

Rigid Body

An idealized structural body assumed to have infinite stiffness, meaning applied forces and moments cause no deformation and only serve to translate or rotate the body.

Deformable Body

A body in which applied forces cause internal points to move relative to one another, altering distance and angular position ($\Delta \theta$) relative to the center of mass.

Stretch Ratio ($\lambda$)

A unitless, geometry-independent metric of deformation defined as the deformed length divided by original length ($\lambda = \frac{L}{L_0}$).

Engineering Strain ($\varepsilon$)

A unitless measure of elongation or shortening normalized to original length, calculated as $\varepsilon = \frac{\Delta L}{L_0} = \lambda - 1$.

True Strain

A strain measure that normalizes change in length by the current, instantaneous length rather than the original length, expressed as $\varepsilon = \frac{L - L_0}{L} = 1 - \frac{1}{\lambda}$.

Shear Strain ($\gamma$)

A measure of angular distortion caused by shear loading, approximated by the tangent of the angular shift ($\gamma = \tan(\theta_2 - \theta_1)$).

Tensile Stress ($\sigma$)

A geometry-independent measure of internal resisting forces distributed across a cut cross section, defined as force divided by cross-sectional area ($\sigma = \frac{F}{A}$).

Constitutive Law

A mathematical relationship between stress and strain that defines the intrinsic mechanical behavior of a material regardless of its geometric configuration.

Elastic Modulus ($E$)

The slope of the linear portion of a stress-strain curve ($E = \frac{\sigma}{\varepsilon}$), representing the intrinsic stiffness of a linearly elastic material.

Yield Strength ($\sigma_y$)

The threshold stress value at which a material ceases to behave elastically and begins to undergo permanent plastic deformation.

0.2% Offset Method

A method to establish yield strength by drawing a line parallel to the linear elastic curve starting at $0.002$ ($0.2\%$) strain and locating its intersection with the stress-strain curve.

Ultimate Tensile Strength ($\sigma_{\text{UTS}}$)

The maximum stress sustained by a material prior to failure, represented as the highest point on the engineering stress-strain curve.

Shear Modulus

A material property analogous to Young's modulus that describes the ratio of shear stress to shear strain.

Poisson's Ratio

A constitutive property quantifying the lateral contraction of a material relative to its longitudinal extension under axial tensile loading.

Linearly Elastic

Behavior wherein stress is directly proportional to strain up to the yield strength, allowing the material to completely return to its original dimensions when unloaded.

Hyperelastic Material

An elastic material whose stress-strain curve exhibits an increasing slope with increasing strain, meaning stiffness increases as it is stretched further (e.g., silicone rubber).

Viscoelastic Material

A material demonstrating mechanical properties that combine characteristics of a viscous liquid and an elastic solid, typical of soft biological tissues.

Permanent Set

The residual plastic deformation remaining in a specimen after it is loaded beyond its yield point and then completely unloaded.

Brittle Material

A material characterized by a steep stress-strain curve and minimal to no plastic region, rupturing abruptly near its ultimate stress without a noticeable slope change.

Ductile Material

A material that undergoes significant, sustained plastic deformation under relatively small increases in stress following its yield point prior to ultimate failure.

Modulus of Resiliency

The recoverable strain energy per unit volume absorbed by a material when deformed up to its yield point, measured as the area under the elastic region of the stress-strain curve.

Modulus of Toughness

The total mechanical energy per unit volume that a material can absorb prior to fracture, represented by the total area under the entire stress-strain curve up to the rupture point.