*2.1. Anisotropic Continuum Plasticity*

In order to describe the elastic-plastic deformation of a material, we introduce the strain tensor *ǫ* that describes the deformation of the material and the stress tensor *σ* that describes the forces acting on the surface of the material. Note that tensorial quantities with rank≥ 1 are typeset in bold letters, whereas scalar quantities are represented by standard characters. In the elastic regime, Hooke's law is used as constitutive relation between stress and strain, such that

$$
\boldsymbol{\sigma} = \mathsf{C} \boldsymbol{\mathfrak{e}}\_{\prime} \tag{1}
$$

where *C* is the fourth-rank elasticity tensor of the material. To describe plastic deformation, the yield function of the material is introduced as

$$f(\mathfrak{o}) = \sigma\_{\mathbf{eq}} - \sigma\_{y'} \tag{2}$$

which takes negative values if the equivalent stress *σ*eq is smaller than the yield strength *σ<sup>y</sup>* of the material, i.e., in the elastic regime. When *f* = 0 plastic yielding sets in, and in case of work hardening, *σ<sup>y</sup>* should be considered as flow stress after this point. Since this work only deals with the onset of plastic yielding, ideal plasticity will be assumed throughout, such that *σ<sup>y</sup>* is a constant, irrespective of the deformation history of the material. Denoting the principal stresses of the stress tensor *σ* as *σ<sup>j</sup>* with (*j* = 1, 2, 3) , the equivalent stress takes the form

$$
\sigma\_{eq}^{\rm L2} = \sqrt{\frac{1}{2} \left[ (\sigma\_1 - \sigma\_2)^2 + (\sigma\_2 - \sigma\_3)^2 + (\sigma\_3 - \sigma\_1)^2 \right]}\,\tag{3}
$$

which—following the definition of von Mises (see, e.g., the translation of the original work by D. H. Delphenich [20])—is based on the second invariant of the stress deviator (J2). In conjunction with the yield function of Equation (2), it describes the onset of plastic yielding for isotropic materials. Note that the formulation in Equation (3) is intrinsically independent of hydrostatic stress components *p* = 1/3Trace(*σ*) and thus does not require to explicitly calculate the deviatoric stress

$$
\sigma' = \sigma - p\mathbf{U},
\tag{4}
$$

where *U* is the unit tensor. By this definition of the equivalent stress, it is inherently assumed that hydrostatic stress components do not affect the plastic flow behavior of the considered material, which is typically fulfilled for metals, but not for polymers or rocks, such that the method formulated here, will mainly apply to metallic materials or, more generally, to materials, where hydrostatic stresses do not influence the plastic behavior.

As described in the introduction, many materials exhibit a directionally dependent yield strength, such that anisotropic flow rules need to be introduced. A first definition of such anisotropic flow rules was introduced by Hill [2], who used a generalized definition of the equivalent stress to achieve a directionally dependent mapping of the equivalent stresses to maintain a constant yield strength. Hence, in this formulation, the anisotropy is considered in the stress rather than in the yield strength. Since the mathematical formulations in this work are purely based on principal stresses, we use a simplified version of the Hill definition and introduce a Hill-like anisotropic definition of the equivalent stress as

$$
\sigma\_{\rm eq} = \sqrt{\frac{1}{2} \left[ H\_1 \left( \sigma\_1 - \sigma\_2 \right)^2 + H\_2 \left( \sigma\_2 - \sigma\_3 \right)^2 + H\_3 \left( \sigma\_3 - \sigma\_1 \right)^2 \right]} \tag{5}
$$

with only three material parameters *H*1, *H*<sup>2</sup> and *H*3, whereas in his original work, Hill introduced three more parameters for an orthotropic material to scale also the shear stress components. Since for orthotropic materials loaded along the main material axes there is no mutual influence of shear and normal components of stress and strain, the formulation introduced here is restricted to loading situations that only produce normal stresses and strains, and where consequently all off-diagonal components of stress and strains tensors remain zero. Furthermore, it is assumed that the loading axes and the main axes of the orthotropic material coincide. Hence, this formalism is currently only valid for a small subset of loading conditions for materials with orthotropic flow anisotropy. The definition of the equivalent stress following Hill can be considered as a generalization of the J2 equivalent stress, because for isotropy, i.e., *H*<sup>1</sup> = *H*<sup>2</sup> = *H*<sup>3</sup> = 1, both definitions are equal.

The restrictions applied in this work, allow the mathematical notation to be simplified by only considering principal stresses. In future work, it is intended to render the formulation more general by exploiting that for any stress state, there exists a coordinate system in which the given stress tensor becomes a diagonal tensor composed of the principal stresses *σ<sup>j</sup>* . This coordinate system is given by the eigenvectors of the stress, representing the principal directions, such that the coordinate system of the original stress tensor—and with it the material axes—can be rotated into the coordinate system of the eigenvectors of the stress tensor. In this orientation the stress tensor becomes a diagonal tensor, and Equation (5) can be evaluated with parameters *H*′ *i* in the rotated state of the material axes.

The thus defined yield function can be used to determine whether a given stress state results in a purely elastic or rather in an elastic-plastic deformation of a material. The condition *f*(*σ*) = 0 relates stresses lying on a specific hyperplane in stress space, the so-called the yield-locus. Since a material does not sustain any stresses larger than the yield stress (for ideal plasticity) or the flow stress (in case of work hardening), acceptable stress states either produce a negative value of the yield function (elasticity) or lie on the yield locus (plasticity), which should be a convex hull of the elastic stress states. Hence, if a predictor step in finite element analysis (FEA) produces a stress outside the yield locus, a plastic strain increment must be calculated that leads again to an accepted stress state on the yield locus. The return mapping algorithm to calculate such strain increments has been described in many text books on continuum plasticity and non-linear FEA, such that here only a very brief summary based on [1] is reproduced. According to the Prandtl–Reuss flow rule, the plastic strain increment for a given time step can be calculated as

$$
\clubsuit\_p = \dot{\lambda} \frac{\partial f}{\partial \sigma} = \dot{\lambda} \mathfrak{n} \, \tag{6}
$$

where *n* is the normal vector to the yield locus, defined by the gradient of the yield function *∂ f* /*∂σ*, and *λ*˙ > 0 is the so-called plastic strain multiplier that can be evaluated as

$$
\dot{\lambda} = \frac{\mathfrak{n} \cdot \mathbf{C} \dot{\mathfrak{e}}}{\mathfrak{n} \cdot \mathbf{C} \mathfrak{n}}.\tag{7}
$$

where *ǫ*˙ is the total strain increment of the FEA predictor step that leads to a stress state outside the yield locus and which is consequently decomposed into the plastic strain increment, given by Equation (6), and the elastic strain increment or stress increment given by

$$
\sigma = \mathbb{C}\_l \mathfrak{e} \tag{8}
$$

with the tangent stiffness tensor

$$\mathbf{C}\_{l} = \mathbf{C} - \frac{\mathbf{C}\mathbf{n} \otimes \mathbf{C}\mathbf{n}}{\mathbf{n} \cdot \mathbf{C}\mathbf{n}} \tag{9}$$

where "⊗" denotes the tensorial product in the form *a<sup>i</sup>* ⊗ *b<sup>j</sup>* = *aib<sup>j</sup>* .

The gradient of the yield function with respect to the principal stresses can be evaluated analytically as

$$\begin{aligned} \frac{\partial f}{\partial \sigma\_1} &= \frac{\partial \sigma\_{eq}}{\partial \sigma\_1} = \frac{(H\_1 + H\_3)\,\sigma\_1 - H\_1\sigma\_2 - H\_3\sigma\_3}{\sigma\_{\text{eq}}}\\ \frac{\partial f}{\partial \sigma\_2} &= \frac{\partial \sigma\_{eq}}{\partial \sigma\_2} = \frac{(H\_2 + H\_1)\,\sigma\_2 - H\_1\sigma\_1 - H\_2\sigma\_3}{\sigma\_{\text{eq}}}\\ \frac{\partial f}{\partial \sigma\_3} &= \frac{\partial \sigma\_{eq}}{\partial \sigma\_3} = \frac{(H\_3 + H\_2)\,\sigma\_3 - H\_3\sigma\_1 - H\_2\sigma\_2}{\sigma\_{\text{eq}}} \end{aligned} \tag{10}$$

Note that in the case of isotropic plasticity (*H*<sup>1</sup> = *H*<sup>2</sup> = *H*<sup>3</sup> = 1), the gradient takes the simple form

$$\frac{\partial f}{\partial \sigma} = 3 \frac{\sigma - p \mathbf{U}}{\sigma\_{\rm eq}} = 3 \frac{\sigma'}{\sigma\_{\rm eq}}.\tag{11}$$

This section served the purpose to introduce the main physical quantities in the notation used in this work. For further details of continuum plasticity or FEA, the reader is referred to standard textbooks, as for example [1]. In the following, the formalism for the data-oriented constitutive model based on a machine learning (ML) yield function is laid out.
