Plasticity (physics)

A yield criterion specifies the admissible elastic stress region, possibly depending on internal hardening variables $q$. Plastic loading occurs on its boundary. The Tresca yield criterion uses maximum principal-stress difference, while the Von Mises yield criterion uses a quadratic deviatoric-stress invariant. Thus a numerical stress difference must be compared to a yield stress using the convention of the selected criterion.

In small-strain elastoplasticity, total infinitesimal strain tensor splits into an elastic part and a generally irreversible plastic part. The elastic constitutive equation determines stress from $e^{\mathrm{e}}$, while a yield condition and a plastic flow law determine the evolution of $e^{\mathrm{p}}$. Removing external load need not remove plastic strain, so using Hooke's law on total strain after yielding is insufficient.

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of $g$, multiplied by a nonnegative plastic multiplier. An associated flow rule takes $g$ equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.

Reduction of an applied load with purely elastic strain increments while accumulated plastic strain is fixed. The superposition principle applies to the increment, so the current stress equals the previous stress plus the linear elasticity response to the load change. The resulting stress must still satisfy the yield surface constraint; reverse yielding ends the purely elastic stage.

Renewed plastic deformation during unloading or reversed loading when the stress reaches the opposite part of the yield surface. For the spherical elastoplastic cavity with maximum pressure $P_m>4{\tau }_Y/3$, elastic unloading first reaches the opposite Tresca yield criterion at the cavity wall after a pressure decrease $8{\tau }_Y/3$. Purely elastic return to zero pressure is possible when $P_m\le 8{\tau }_Y/3$, with equality reaching the yield surface exactly at the endpoint.

The Tresca yield criterion bounds the largest resolved shear stress, expressed through the principal stresses. In spherical symmetry, the two tangential principal stresses coincide, so $f=|{\tau }_{rr}-{\tau }_{\theta \theta }|/2$. Here ${\tau }_Y$ is a shear yield stress; the corresponding uniaxial yield strength is $2{\tau }_Y$.

Localization of tensile deformation into a narrowing region of a specimen. The onset depends on the competition between declining cross-sectional area and work hardening.

Plastic deformation below the temperature range for substantial concurrent recrystallization. It changes dislocation structure and texture; irradiation and later heating can modify the resulting microstructure.

An increase in resistance to further plastic deformation after deformation has changed the microstructure. Increasing dislocation density and interactions can impede subsequent dislocation glide.

Increase in the resistance of one slip system caused by plastic shear on another. It can shift the onset of secondary slip away from a purely geometric equal-Schmid-factor prediction.

Resistance to dislocation glide produced by intersections with other dislocations. Multiple active slip systems can form junctions and increase the density of obstacles. This is one mechanism of work hardening.

A yield point marks onset of permanent deformation under the stated experimental criterion. A specified offset proof stress is an operational measure when a sharp yield transition is absent.

An offset proof stress is defined by the intersection of a loading curve with a line parallel to its initial linear elasticity response but shifted by a specified engineering strain. A 0.2 percent convention uses the line $\sigma =E(\epsilon -0.002)$, with Young's modulus $E$.

The Von Mises yield criterion depends only on the second invariant of deviatoric stress. With shear yield stress $k$, its surface is ${\sigma }^{\prime }:{\sigma }^{\prime }=2k^2$; the uniaxial yield stress is $\sqrt{3}k$. An associated flow rule gives plastic strain rate proportional to ${\sigma }^{\prime }$, up to a positive normalization of the plastic multiplier.

Plastic strain rate is normal to the yield surface in the work-conjugate stress variables. With symmetric stress variations, the pairing is $D:\delta \sigma$; in two dimensions its off-diagonal contribution is $2D_{12}\delta {\sigma }_{12}$. This factor is essential when changing to reduced stress coordinates.

A perfectly plastic material has a yield surface that does not change with accumulated plastic deformation. This removes hardening while retaining the distinction between elastic and plastic loading.

A spherical cavity of fixed reference radius $a$ in an infinite solid with perfect plasticity, the Tresca yield criterion, and zero far-field stress first yields under internal pressure $P_c$. In the plastic region $a<r<s$, radial force balance and ${\tau }_{\theta \theta }-{\tau }_{rr}=2{\tau }_Y$ give ${\tau }_{rr}=-P+4{\tau }_Y\log (r/a)$. Matching to the outer linear elasticity solution, where ${\tau }_{rr}(s)=-P_c$, gives the displayed plastic-front radius. These are small-strain reference-geometry formulae.

In antiplane perfect plasticity, velocity $w=v_3$ and shear stresses $a={\sigma }_{13},b={\sigma }_{23}$ depend only on $x_1,x_2$. Force balance is $a_{,1}+b_{,2}=0$, the fixed yield surface is $f(a,b)=k$, and the associated flow rule is $\nabla w=2\dot{\lambda }(f_a,f_b)$ with $\dot{\lambda }\ge 0$. Differentiating the yield equation in each coordinate and combining it with equilibrium shows that $a,b,w$ are constant along characteristic curves tangent to $(-f_b,f_a)$. For example $-f_b{a}_{,1}+f_a{a}_{,2}=f_b{b}_{,2}+f_a{a}_{,2}=0$. Nonzero regular yield gradient is needed for a characteristic direction.

A centred fan consists of straight characteristic curves emanating from one point. Stress and antiplane velocity are constant on each ray and depend on the polar angle $\varphi$. The radial tangent condition is $(\cos \varphi ,\sin \varphi )\parallel (-f_b,f_a)$, so $f_a\cos \varphi +f_b\sin \varphi =0$. A boundary ray carrying a traction-free crack-face stress $b=0$ has slope $\tan {\varphi }_b=-f_a/f_b$, interpreted as vertical when $f_b=0$. For the elliptic yield condition $a^2/A^2+b^2/B^2=1$, a forward fan has $a=-A^2\sin \varphi /H$, $b=B^2\cos \varphi /H$, $H=(A^2{\sin }^2\varphi +B^2{\cos }^2\varphi {)}^{1/2}$. The boundary rays are vertical; the adjacent rear regions have constant stresses $(\mp A,0)$. Velocity is a function of angle whose magnitude requires further loading data.

In plane strain perfect plasticity with ${\sigma }_{11}=-p+k\sin 2\varphi$, ${\sigma }_{22}=-p-k\sin 2\varphi$ and ${\sigma }_{12}=-k\cos 2\varphi$, equilibrium givesThe slip line tangent directions $(\cos \varphi ,\sin \varphi )$ and $(-\sin \varphi ,\cos \varphi )$ are eigenvectors with eigenvalues $+1,-1$. Taking directional derivatives proves the two Hencky stress relations. These sign choices must be changed consistently if the shear-stress convention changes.

For a symmetric reentrant notch opening towards negative $x_1$, let its semi-angle be $0\le \gamma \le \pi /2$. On the upper free face, tensile tangential stress selects $p=-k$, $\varphi =\pi /4-\gamma$. The adjacent centred fan has radial $\beta$ lines and circular $\alpha$ lines: $\varphi =\vartheta -\pi /2$, $p=-k+2k[\vartheta -(3\pi /4-\gamma )]$, for $\pi /4\le \vartheta \le 3\pi /4-\gamma$. It joins a constant stress region containing the forward axis, where $\varphi =-\pi /4$ and $p=-k(1+\pi -2\gamma )$. Therefore ${\sigma }_{22}=k(2+\pi -2\gamma )$ in that fully plastic notch-tip region. This is a local stress field, not a claim that an arbitrary specimen is plastically yielded everywhere.

For an associated flow rule in the reduced plane-strain yield coordinates, the rate-of-strain tensor has zero extension along both slip line directions. Write velocity as $u{e}_{\alpha }+v{e}_{\beta }$, with the local orthonormal axes turning through angle $\varphi$. Differentiating along an $\alpha$ curve gives longitudinal extension $du-vd\varphi$; along a $\beta$ curve it gives $dv+ud\varphi$. Setting these extensions to zero proves the relations for the corresponding anisotropic yield surface parameterization.

Parameterize a fixed yield surface by stress-space arc length $l$ with ${\xi }_{,l}=-\cos 2\varphi$ and ${\tau }_{,l}=-\sin 2\varphi$. For mean normal stress $\sigma$, force balance gives $\nabla \sigma =M\nabla l$, where $M=\left(\begin{array}{cc}\cos 2\varphi & \sin 2\varphi \\ \sin 2\varphi & -\cos 2\varphi \end{array}\right)$. Its eigenvectors along angles $\varphi$ and $\varphi +\pi /2$ have eigenvalues $+1$ and $-1$, proving the displayed slip line invariants.

A slip line is a characteristic curve of a yielded plane-strain plastic equilibrium system. The two orthogonal families carry stress and velocity compatibility relations. Their directions need not agree with principal stress directions.