# Background

Often in geological settings, rocks are deposited in layers, and between the layers there are joints, which are usually weak, and which we call "weak planes" here. Upon applying stresses/strains, these joints often fail by opening (Weak-plane tensile failure), or by shearing (Weak-plane shear failure). Here we describe Weak-plane shear plasticity, which is non-associative and includes hardening/softening.

# Definition

Define a vector, $$n$$$, which is normal to the weak plane. The user may choose two options: 1. This vector rotates with the large deformations 2. This vector remains fixed in an external reference frame Usually the first choice would be made. Denote the so-called "cohesion" by $$C$$$, the "friction angle" by $$\phi$$$, and the "dilation angle" by $$\psi$$$. These may be constant, or governed by a Hardening or Softening law

At the start of each "time-step" (increment), rotate the quad-point's reference frame so that $$n$$aligns with the $$z$$ axis. Denote the rotated stress by $$\tilde{\sigma}$$$. Then the shear yield function is f(\phi) = \sqrt{\tilde{\sigma}_{zx}^{2} + \tilde{\sigma}_{zy}^{2}} + \tilde{\sigma}_{zz}\tan\phi - C \ , and the flow potential is r_{ij} = \frac{\partial f(\psi)}{\partial \tilde{\sigma}_{ij}} \ . Notice the appearance of the dilation angle $$\psi$$$ in this formula. Associative plasticity is obtained for $$\psi=\phi$$$. The hardening law is defined via h = -1 \ , so that $$q = q^{\mathrm{old}} + \gamma$$$, ie, $$q$$$is the accumulated sum of $$\gamma$$$ over all "time steps".

# The test suite

A number of tests have been implemented demonstrating that the MOOSE code correctly implements the return-map algorithm for weak-plane shear plasticity, both for small strains and large strains. These are:

Weak-plane shear envelope for the smoothed and unsmoothed case. The MOOSE results were obtained by applying a sequence of deformations to a single element and letting the stress return-map to the yield surface.

1. small_deform1. Denote the so-called "normal stress", $$\tilde{\sigma}_{zz}$$$by $$N$$$ for ease of notation. The parameters in this test are $$C=1$$$, $$\tan\phi=1/2$$$, and perfect plasticity is used, so that the cone's tip lies at $$N=2$$$. Poisson's ratio is zero, and Lame $$\mu = 10^6$$$. Apply a deformation so the trial stress has $$\tau^{\mathrm{trial}}=10$$$and $$N^{\mathrm{trial}}=2$$$. The return-map must then solve $$f=0$$$, and \tau = \tau^{\mathrm{trial}} - \frac{1}{2}\mu\gamma and N = N^{\mathrm{trial}} - \mu\tan(\psi)\gamma Finally, the dilation angle is chosen to be $$\tan\psi = 2/18$$$. Then the returned value should be $$\tau=1$$$and $$N=0$$$.

2. small_deform2. A sequence of boundary displacements are applied to a single element. MOOSE uses the return-map algorithm to map out the weak-plane shear envelope (perfect plasticity is used). The result is shown to the right, and clearly MOOSE returns correctly to the yield surface.

3. small_deform3. Ten-thousand random small deformations are applied to a single element using perfect plasticity. It is checked that MOOSE returns to the yield surface each time.

4. small_deform4. A sequence of boundary displacements are applied to a single element, and 'cap' smoothing is used. MOOSE uses the return-map algorithm to map out the weak-plane shear envelope (perfect plasticity is used). The result is shown to the right, and clearly MOOSE returns correctly to the yield surface.

5. large_deform1. The normal direction is chosen $$n=(0,0,1)$$$, and it rotates with the mesh. A single element is rotated through 90 degrees about the $$x$$$ axis, and then stretched in $$z$$$direction. No plastic deformation is observed, in agreement with expectations. 6. large_deform2. The normal direction is chosen $$n=(0,0,1)$$$, and it rotates with the mesh. A single element is rotated through 45 degrees about the $$x$$$axis, and then stretched in the $$y=z$$$ direction. This should create a pure tensile load ($$\tau=0$$$), which should return to the yield surface. The parameters chosen are $$C=10^6$$$, $$\tan\phi=1$$$, and $$a=5\times 10^5$$$ (smoothing parameter), so the tip of the smoothed cone is at $$5\times 10^5$$$(perfect plasticity is used). The result of the stretch should be \sigma_{yy} = \sigma_{yz} = \sigma_{zz} = 2.5\times 10^5 \ , and this is indeed the result obtained by MOOSE. 7. large_deform3. Ten-thousand random large deformations are applied to a single element with $$n$$$ rotating with the mesh. Perfect plasticity is used. It is checked that MOOSE returns to the yield surface each time.

8. large_deform4. Ten-thousand random large deformations are applied to a single element with $$n$$$rotating with the mesh. Cap smoothing is used. Perfect plasticity is used. It is checked that MOOSE returns to the yield surface each time. 9. small_deform_harden1. Repeated stretches in the $$z$$$ direction are applied to a single element (with $$n=(0,0,1)$$$). Exponential hardening is used: $$C_{0}=1000$$$, $$C_{\mathrm{res}}=2000$$$, and $$\zeta_{C}=40000$$$, and the smoothing is done with $$a=500$$$. The successive return-maps to the yield surface induce hardening of the cohesion, which agrees with the expected result as shown in the Figure. 10. small_deform_harden2. Apply repeated stretches and shears to map out the yield function in a similar way to small_deform2, but this time use exponential hardening/softening. The following parameters are used $$C_{0}=1000$$$, $$C_{\mathrm{res}}=700$$$, and $$\psi_{0}=5$$$, $$\psi_{\mathrm{res}}=1$$$, and $$\phi_{0}=45$$$, and $$\phi_{\mathrm{res}}=30$$$. Smoothing is done with $$a=500$$$. The Figure shows the initial yield surface, and the fully developed yield surface (ie when $$\zeta q\gg 1$$$). 11. small_deform_harden3. This is similar to small_deform_harden1, but with exponential hardening occurring through an evolving friction angle: $$\phi_{0}=45$$$, $$\phi_{\mathrm{res}}=30$$$, $$\zeta_{\phi}=40000$$$. The results are shown in the Figure.

12. small_deform_harden4. The same as small_deform_harden1, but with cubic hardening.

13. large_deform_harden3. The idea is the same as large_deform3, but with exponential hardening/softening: $$C_{0}=1000$$$, $$C_{\mathrm{res}}=0$$$, $$\phi_{0}=45$$$, $$\phi_{\mathrm{res}}=10$$$, $$\psi_{0}=5$$$, $$\psi_{\mathrm{res}}=2$$$.

Hardening of cohesion. Results from small_deform_harden1

Original yield surface, and fully developed yield surface (when $$\zeta q \gg 1$$$for all $$\zeta$$$). Results from small_deform_harden2.

Evolution of the friction angle causes evolution of the yield function. Results from small_deform_harden3.

Yield function for the 'cap' type of smoothing.

Hardening of cohesion with cubic hardening. Results from small_deform_harden4