2. Benchmark solution

2.1. Calculation method

In the following, all the quantities defined in the initial configuration will be written in capital letters. The quantities defined in the deformed configuration will be written in lower case.

To establish the analytical solution, we consider a sphere with initial radii inside \(\text{A}\) and outside \(\text{B}\). The material is elastoplastic with perfect plasticity. The plastic flow is normal and associated with the von Mises criterion, elastic limit \({\sigma }_{y}\). Elasticity is isotropic defined by the compressibility and shear modules, \(K\) and \(\mu\). The sphere is subject to internal pressure \(P\).

Given the spherical symmetry of the geometry and the load, we are looking for a solution that has the same invariance properties, which excludes the search for bifurcated solutions. The solution only depends on the distance to the center of the sphere noted \(R\) in the initial configuration and \(r\) in the deformed configuration. The displacement is purely radial: \(\vec{u}(r)=u(r){\vec{e}}_{r}\), where \(({\vec{e}}_{r},{\vec{e}}_{\theta },{\vec{e}}_{\phi })\) is the orthonormal base associated with the spherical coordinate system. We will therefore use the definition of the appropriate operators \(\text{grad}\), \(\text{div}\),…. The stress tensor is expressed as follows:

(2.1)\[ \ sigma = {\ sigma} _ {\ mathit {rr}} (r) {\ vec {rr}} (r) {\ vec {e}} _ {r} + {\ sigma} _ {\ sigma} _ {\ theta\ theta}} (r) (r) (r) ({\ vec {e}} (r) (r) ({\ vec {e}} (r) (r) (r) (r) ({\ vec {e}}) _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} _ {\ vec {e}} e}} _ {\ phi}\ otimes {\ vec {e}} _ {\ phi})\]

As far as displacement is concerned, its expression in the reference configuration is: \(\vec{u}(\vec{X})=U(r){\vec{e}}_{r}\). We can notice that, by definition, we have the relationships:

\(r=R+U(R)\) and \(u(r)=U(R)\)

We deduce the following formula for derivatives

(2.2)\[ \ frac {\ partial U} {\ partial R} =\ frac {\ partial R} {\ partial u} {\ partial r}\ frac {\ partial r} {\ partial R} =\ frac {\ partial u} {\ partial r} {\ partial r}\ left (1+\ frac {\ partial r}}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial U}\ partial r}\ left (1+\ frac {\ partial ac {\ frac {\ partial u} {\ partial r}} {\ partial r}} {\ left (1-\ frac {\ partial u} {\ partial r}\ right)}\]

The transformation gradient is calculated:

(2.3)\[ F=\ left (1+\ frac {\ partial U} {\ partial U} {\ partial R}\ right) {\ vec {e}} _ {r}\ otimes {\ vec {e}} _ {r} +\ left (1+\ frac {U} {U} {R}}\ right) ({\ vec {R}}\ right) ({\ vec {e}}\ right) ({\ vec {e}}} _ {\ theta}\ otimes {\ vec {e}} _ {r}} +\ left (1+\ frac {U} {R} {R}\ right) ({\ vec {e}}\ right) ({\ vec {e}}\ right) ({\ vec {e}}\ right) ({\ vec {e}}} _} {\ vec {e}} _ {\ phi}\ otimes {\ vec {e}} _ {\ phi})\]

Note:

\(F\) admits constant natural directions during the transformation.

The logarithmic deformation then simply admits as an expression:

\[\]

: label: eq-4

E=mathrm {ln}left (1+frac {partial U} {partial U} {partial R}right) {vec {e}} _ {e}} _ {r} _ {r} +mathrm {ln}left (1+frac {N}right) ({vec {e}}} _ {r}} _ {r}} _ {r}} _ {r}} +mathrm {ln}\ left (1+frac {U} {R}right) ({vec {e}}} _ {r}} _ {r}} _ {r}} _ {r}} _ {r}} +mathrm {ln}}left (1+frac {U} {R}right otimes {vec {e}} _ {theta} {with {e}} _ {phi}otimes {vec {e}}} _ {theta}} _ {phi})

Note:

In this particular case, the deformation rate \(\dot{E}\) coincides with the Eulerian deformation ratio \(D\) , so that the stress tensor \(T\) associated with logarithmic deformations is equal to the Kirchhoff content \(\tau\) . It is a favorable situation to demonstrate an analytical solution.

In the deformed configuration, deformation \(e(\vec{x})=E(\vec{X})\) is expressed:

\[\]

: label: eq-5

e=-mathrm {ln}left (1-frac {partial u} {partial u} {partial r}right) {vec {e}} _ {r}otimes {vec {e}} _ {r}} _ {r} +r} +left (vec {e}} +left (vec {e}}} _ {theta}otimes {vec {e}}} _ {theta} {vec {e}} _ {phi}otimes {vec {e}}} _ {phi})

It is also useful to express the Jacobian of the transformation and its logarithm in the deformed configuration:

(2.4)\[ j=\ text {det} F=\ frac {1} {\ left (1-\ frac {\ partial u} {\ partial r}\ right)} {\ left (\ frac {r} {r} {r} {r-u}\ right)} ^ {2}\]

The equilibrium equations can also be expressed in the deformed configuration as a function of the Kirchhoff tensor \(\tau =j\sigma\):

(2.5)\[ {\ text {div}} _ {x}\ left (\ frac {\ tau} {j}\ right) =0\ iff\ frac {\ partial {\ tau} _ {\ mathit {rr}}}}} {\ partial r}}} {\ partial r}} - {\ partial r} - {\ tau} _ {\ tau} _ {\ tau} _ {\ mathit {rr}}}\ frac {\ partial {\ tau}} _ {\ mathit {rr}} _ {\ mathit {rr}}} {\ partial r}}} {\ partial r}} - {\ partial r} - {\ tau} - {\ tau} _ {\ tau} _ {\ mathit {rr}} {\ partial} j} {\ partial r}}} {\ partial r}} - {2} {r} ({\ tau} _ {\ mathit {rr}}} - {\ tau} _ {\ theta\ theta}) =0\]

In addition, we have the boundary conditions:

(2.6)\[ {\ tau} _ {\ mathit {rr}} (b) =0\]

Where \(a\) and \(b\) are the inner and outer radii of the deformed sphere. Given spherical symmetry, it is equivalent to imposing internal pressure or radial displacement on the inner skin. It is preferable to control the calculation while moving, so that the second boundary condition is replaced by:

(2.7)\[ u (a) = {U} ^ {\ mathit {imp}}\]

In terms of behavior, the hydrostatic part is purely elastic:

(2.8)\[ \ text {tr}\ tau =3K\ text {tr} e\ iff 3K\ mathrm {ln} j= {\ tau} _ {\ mathit {rr}}} +2 {\ tau}} +2 {\ tau} _ {\ theta\ theta}\]

For the deviatoric part, we have the classical relationships of von Mises plasticity:

(2.9)\[ {\ tau} _ {\ mathit {eq}}\ le {\ sigma} _ {y};\ dot {p}\ ge 0;\ dot {p} ({\ tau} _ {\ tau} _ {\ mathit {eq}}} - {\ sigma} _ {y}) =0\]

(2.10)\[ {\ tau} ^ {D} =2\ mu ({e} ^ {D} + {e} ^ {p})\]

In terms of structural response, a scenario is expected in which the plastic zone grows progressively from the inside to the outside of the sphere. When it reaches the outer wall, all the points of the sphere are in a plastic regime, i.e.:

(2.11)\[ {\ tau} _ {\ mathit {eq}} = {\ sigma}} = {\ sigma} _ {\ sigma} _ {\ theta\ theta} - {\ tau} _ {\ mathit {rr}} = {\ sigma} _ {y}\]

In addition, at the moment when the external wall is affected, the plastic deformation is still zero. So we have:

(2.12)\[ {e} ^ {D} (b) =\ frac {1} {2\ mu} {2\ mu} {\ mu} {\ tau} (b)\ iff\ frac {({\ tau} _ {\ mathit {rr}}} (b) - {\ tau}} (b) - {\ tau}} {\ tau}} {\ tau}} (b)} {\ mathrm {rr}}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau}} (b) - {\ tau} ac {1-\ frac {\ partial u} {\ partial u} {\ partial r}} {r-u}\ right)\ mid} _ {b} +\ mathrm {ln} b\]

When the plastic zone opens onto the outer wall, one of the two components of the stress tensor is known at every point via (13). The equilibrium relationship (7) as well as the spherical part of the behavior (10) then make it possible to fully determine the stress field, as well as the Jacobian of the transformation. In fact, unlike the case of small deformations, these two equations are coupled. They are written:

(2.13)\[\begin{split} \ {\ begin {array} {c}\ frac {\ partial {\ tau} _ {\ mathit {\ tau}}} {\ partial r} - {\ tau} _ {\ mathit {rr}}\ frac {\ partial\ partial\ mathrm {ln} j} {\ partial r}} {\ partial r}}} {\ partial r}} - {\ tau} _ {\ mathit {rr}}} - {\ tau} _ {\ theta\ theta}) =0\\\ mathrm {3K}\ mathrm {ln} j= {\ tau} _ {\ mathit {rr}}} +2 {\ tau}} +2 {\ tau}} +2 {\ tau} _ {\ tau} _ {\ tau} _ {\ theta\ theta}}\ end {array}\end{split}\]

Substituting the definition of \(\mathrm{ln}j\) into the first equation of (15), we get:

(2.14)\[ \ frac {\ partial} {\ partial r}\ left ({\ tau} _ {\ mathit {rr}}} -\ frac {{\ tau} _ {\ mathit {rr}}} ^ {2}}} ^ {2}}}} {\ left) {\ mathrm {2K}}}\ right) =\ frac {2} {r} {\ sigma} _ {y}\]

This equation is easily integrated by taking into account the boundary condition (8)

(2.15)\[ {\ tau} _ {\ mathit {rr}}} -\ frac {{\ tau}} _ {\ mathit {rr}} ^ {2}} {\ mathrm {2K}}} =2 {\ sigma}} _ {y}\ y}\ y}\ y}\ y}\ y}\ y}\ y}\ y}\ y}\ mathrm {ln}\ left (\ frac {r} {b}\ right)\]

The constraint field can be deduced by solving this second-order equation. The choice of the root is determined by the fact that it is a compressive stress, and therefore a negative one. The volume change field \(\mathrm{ln}j\) is then deduced from this through (15):

(2.16)\[\begin{split} \ {\ begin {array} {c} {\ tau} _ {\ tau} _ {\ mathit {rr}} (r) =K-\ sqrt {{K} ^ {2} -\ mathrm {4K} {\ sigma} _ {\ sigma} _ {y} _ {y}\ y}\\\ mathrm {\ sigma} _ {y} _ {y}\\\ mathrm {\ sigma} _ {y} _ {y}\\\ mathrm {_} _ {y}\\ mathrm {_} _ {y}\\ mathrm {ln} _ {y}\\ mathrm {ln} _ {y}\\ mathrm {ln} _ {y}\\ mathrm {ln} _ {y}\\\ mathrm {ln}\ sqrt {1-4\ frac {{\ sigma} _ {y}}} {K}\ mathrm {ln}\ left (\ frac {r} {b}\ right)}} +\ frac {2 {\ sigma} _ {\ sigma} _ {y}} {y}} {\ mathrm {3K}}\ end {array}}\end{split}\]

Moreover, the kinematic relationship (6) can be rewritten:

(2.17)\[ \ frac {\ partial} {\ partial r}\ left ({(r-u)}} ^ {3}\ right) =\ frac {3 {r} ^ {2}} {\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {ln} j}\]

Taking into account definition \(b-u(b)=B\), we deduce by integration:

(2.18)\[ {(r-u)} ^ {3} = {B} ^ {2} - {\ int} - {\ int} _ {r} ^ {b}\ frac {3 {\ rho} ^ {2}}} {\ mathrm {exp}}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {exp}\ mathrm {\]

The primitive in (20) should be computed numerically.

We can also rely on the relationship (19) to simply express the deformed ray in \(b\). In fact, in \(r=b\), the volume change is equal to:

\[\]

: label: eq-21

mathrm {ln} j (b) =frac {2 {sigma} _ {y}} {mathrm {3K}}}

Equation (19) is written in \(r=b\):

(2.19)\[ \ left (1-\ frac {\ partial u} {\ partial u} {\ partial r}\ right) =\ frac {{b} ^ {2}} {{B} ^ {2}}\ mathrm {exp}\ left (-\ frac {2 {r}}\ left (-\ frac {2 {\ sigma}}\ right)\]

The continuity condition (14) then makes it possible to fix the radius of the deformed sphere because it is still expressed:

(2.20)\[ \ left (1-\ frac {\ partial u} {\ partial u} {\ partial r}\ right) = {\ left (\ frac {b} {B}\ right)} ^ {-1}\ mathrm {exp}\ left\ left (\ frac {{\ sigma}} _ {y}} {2\ mu}\ right)\]

From (22) and (23), the deformed external radius is deduced:

(2.21)\[ \ frac {b} {B} =\ mathrm {exp}\ left (\ frac {{\ sigma} _ {y}} {3}\ left (\ frac {1} {2\ mu} {2\ mu} +\ frac {2}} {\ mathrm {3K}}\ left)\ right)\ right)\]

Knowledge of deformation and stress makes it possible in turn to determine the cumulative plastic deformation field \(p\). Indeed, its evolution is governed by the tensor equation (12). More precisely, it turns out that the direction \({\tau }^{D}/{\tau }_{\mathit{eq}}\) is constant, which makes it possible to simply integrate the plastic deformation:

(2.22)\[ {e} ^ {p} =p\ left ({\ vec {e}}} _ {r}\ otimes {\ vec {e}} _ {r} -\ frac {1} {2} {\ vec {e}}} _ {\ vec {e}} _ {\ theta}} _ {\ vec {e}} _ {\ vec {e}}} _ {\ phi}\ otimes {\ vec {e}} _ {\ phi}\ right)\]

Then by substituting this expression into the first equation of (12), we deduce:

(2.23)\[ p (r) =\ frac {2} {3} ({e}} _ {\ theta\ theta}} - {e} _ {\ mathit {rr}}) -\ frac {{\ sigma} _ {y}}} {y}} {3}} {3} =\ frac {2} {3}}\ left (\ mathrm {ln} r-\ frac {{\ sigma} _ {y}} {2\ mu} +\ mathrm {ln}\ frac {1-\ frac {\ frac {\ partial u} {\ partial r}} {r-u}\ right)\]

Where the expression for the displacement field is now known from (20).

Finally, the inner boundary condition makes it possible to determine the critical load level for which the plastic zone reaches the outer wall. In fact, the distorted inner radius is given implicitly by the relationship (20) expressed precisely in \(r=a\):

\[\]

: label: eq-27

{int} _ {r} ^ {b}frac {3 {rho} ^ {2}} {mathrm {exp}mathrm {ln} j (rho)} drho} drho = {B} ^ {3} - {A} ^ {3}}

2.2. Reference quantities and results

The reference quantities are the stress trace and the cumulative plastic deformation in deformed configuration for the Gauss points with the smallest and the largest r.

2.3. Uncertainties about the solution

Since the solution is analytical, there is no uncertainty.

2.4. Bibliographical references

1. 5. LORENTZ, « Dualization of conditions of almost incompressibility », Internal document EDF R&D, 2011.