Constraints on the applicability range of pressure-sensitive yield/failure criteria: strong orthotropy or transverse isotropy

Yield/failure initiation criteria discussed in this paper account for the three following effects: the hydrostatic pressure dependence, the tension/compression asymmetry, and isotropic or anisotropic material response. For isotropic materials, criteria accounting for pressure/compression asymmetry (strength differential effect) have to include all three stress invariants (Iyer and Lissenden in Int J Plast 19:2055–2081, 2003; Gao et al. in Int J Plast 27:217–231, 2011; Yoon et al. in Int J Plast 56:184–202, 2014; Coulomb–Mohr’s, cf. Chen and Han in Plasticity for structural engineers. Springer, Berlin, 1995 criteria). In narrower case when only pressure sensitivity is accounted for, rotationally symmetric surfaces independent of the third invariant are considered and broadly discussed (Burzyński in study on strength hypotheses (in Polish). Akad Nauk Tech Lwów, 1928; Drucker and Prager in Q Appl Math 10:157–165, 1952 criteria). For anisotropic materials, the explicit formulation based on either all three common invariants (Goldenblat and Kopnov in Stroit Mekh 307–319, 1966; Kowalsky et al. in Comput Mater Sci 16:81–88, 1999) or first and second common invariants (extended von Mises–type Tsai–Wu’s criterion in Int J NumerMethods Eng 38:2083–2088, 1971) is addressed especially for the case of transverse isotropy, when difference between tetragonal versus hexagonal symmetry is highlighted. The classical Tsai and Wu criterion involves Hill’s type fourth-rank tensor inheriting a possibility of convexity loss in case of strong orthotropy, as discussed by Ganczarski and Skrzypek (Acta Mech 225:2563–2582, 2014). In order to overcome this defect, in the present paper the new Mises-based Tsai–Wu’s criterion is proposed and exemplary implemented for the columnar ice. A mixed way to formulate pressure-sensitive tension/compression asymmetric failure criteria-capable of describing fully distorted limit surfaces, which are based on both all stress invariants and the second common invariant (Khan and Liu in Int J Plast 38:14–26, 2012; Yoon et al. in Int J Plast 56:184–202, 2014), is revised and addressed to orthotropic materials for which the fourth-order linear transformation tensors are used to achieve extension of the isotropic criterion.

Since the present paper is an extension of previous work [16] for a case of limit surfaces of pressuresensitive materials, it is essential to invoke the general formulation of limit surface based on the Goldenblat and Kopnov concept of common invariants.
In a most general case of both elastic and plastic material anisotropy, extension of the isotropic plastic yield initiation criterion to the anisotropic flow by the use of common invariants of the stress tensor and of the structural tensors of plastic anisotropy (cf. Hill [24], Sayir [50], Betten [3],Życzkowski [62]), can be shown in a general fashion (2) : σ , σ : (4) : σ , σ : (6) : σ : σ , . . . = 0 where Einstein's summation holds. Note that the general form Eq. (1) uses unspecified representation of an unlimited number of common invariants. In such a case, initiation of plastic flow is governed by the structural tensors of plastic anisotropy of even-ranks: (0) = Π, (2) , (4) , (6) , . . ., etc. Equation (1) owns a general representation, but its practical identification is limited by a large number of required material tests and, additionally, because the components of the structural tensors are temperature dependent, which makes identification much more complicated (cf. e.g. Herakovich and Aboudi [23], Tamma and Avila [56]). Hence, the general form (1) is usually specialized for engineering needs.
Equation (3), when limited only to three common invariants of the stress tensor σ and structural anisotropy tensors of even orders: 2 nd π, 4 th and 6 th I¯I, is not the most general one, in the meaning of the representation theorems, which determine the most general irreducible representation of the scalar and tensor functions that satisfy the invariance with respect to a change of coordinates and material symmetry properties (cf. e.g. Spencer [52], Rymarz [49], Rogers [48]). However, 2nd, 4th and 6th order structural anisotropy tensors, which are used in (3), are found satisfactory for describing fundamental transformation modes of limit surfaces caused by plastic or damage processes, namely: isotropic change of size, kinematic translation and rotation, as well as surface distortion (cf. Kowalsky et al. [32], Betten [3]). Goldenblat and Kopnov's equation (3) is quite general, too. Hence, for some engineering applications, its further reduction is performed. Namely, assuming α = β = γ = 1, a following format of anisotropic failure initiation criterion is obtained: π : σ + σ : : σ + σ : I¯I : σ : σ − 1 = 0.
When compared to the analogous polynomial format used as yield criterion [16] in Eq. (4), all three terms are saved, and the first invariant π : σ plays an essential role when pressure-sensitive materials are considered.
Although the general format of the failure initiation criterion (4) is analogous to the previously introduced criterion of the yield initiation [16], further reduction of this equation to be applicable for brittle materials has to be performed applying different assumptions from those used for ductile materials. Plastic yield initiation in metallic polycrystalline materials is traditionally characterized by the following features (compare [16]): -yield initiation is usually hydrostatic pressure insensitive (independent of the first common invariant), -the criterion of yield initiation exhibits symmetry with respect to tension and compression (no strength differential effect), -yield surface has to be convex (Drucker's postulate).
On the other hand, failure initiation in some metallic or non-metallic materials is commonly described including the more complex behaviour: -failure initiation is hydrostatic pressure sensitive, -the criterion of failure initiation exhibits asymmetry with respect to tension and compression (also called strength differential effect), since for the majority of brittle materials strength resistance for compression is frequently much higher than for tension (essential strength differential effect), -the failure surface has to be convex (positive definiteness of tangent stiffness matrix in Sylvester's sense-see "Appendix D").
The above features are briefly discussed by Ganczarski and Skrzypek in [18]. In the light of experimental observation of metal alloys, a distinction between the plastic mechanism and the brittle mechanism of failure initiation may not be essential and justified.
Nevertheless, it should be pointed out that in all described cases of physically different coupled mechanisms the limit surface of yield and/or failure initiation must definitely satisfy the convexity requirement in the sense of either Drucker's or Sylvester's material stability postulates.
2 Remarks on isotropic yield/failure criteria accounting for the pressure sensitivity and strength differential effect In general case of the yield/failure initiation in isotropic materials, the equation of the limit surface in terms of the three basic stress invariants J 1σ , J 2σ , J 3σ takes a form where symbol k i stands for a set of material constants, for instance k t , k c , k s . This equation describes the general class of isotropic hydrostatic pressure-sensitive materials; see items C1-C5 in Table 7.
In the narrower case of failure in isotropic materials, the equation of the limit surface is commonly written in terms of the following three invariants: the first stress tensor invariant J 1σ , and the second J 2s , and the third J 3s stress deviator invariants The advantage of Eq. (6) is that it separates the hydrostatic stress J 1σ from the influence of deviatoric stresses expressed by J 2s and J 3s . Such surface is no longer cylindrical, and hence it naturally exhibits tension/compression asymmetry. All limit surfaces belonging to the class considered exhibit certain sectorial symmetry with respect to the hydrostatic axis (see discussion in [16]); however, only in a particular case of independence of the third invariant J 3s (or θ ), it is fully rotationally symmetric. The Ottosen and Ristinmaa [40] mixed format separates the influence of the hydrostatic pressure J 1σ from the deviatoric stress represented by J 2s and cos 3θ , where the following definitions hold: The invariants J 1σ , J 2s and cos 3θ have clear interpretation: J 1σ tells about the hydrostatic pressure sensitivity (non-cylindrical limit surface), J 2s represents the distance of the point at the deviatoric limit curve (magnitude of the deviatoric stress), and cos 3θ informs about the influence of the direction of deviatoric stress (asymmetry in so-called strength differential materials), cf. Ottosen and Ristinmaa [40]. Applying consistently a system of the first stress invariant and the second and the third stress deviator invariants, Eq. (7) can also be written as (6). The other stress invariant systems are broadly discussed by Haigh [22], Westergaard [59], Novoshilov [39],Życzkowski [61], and summarized by Altenbach et al. [2] in a recent important monograph. In the last one monograph, the authors provide a complete review and comparison of a variety of phenomenological yield/failure pressure-insensitive versus pressure-sensitive isotropic criteria. In particular for novel materials, the following phenomena are taken into account: tension/compression asymmetry, also called strength differential effect, hydrostatic pressure sensitivity, the Poynting-Swift effect as well as the Kelvin effect. In contrast to conventional material behaviour, described by tensorial linear equations, the above phenomena require tensorial nonlinear equations; therefore, they may be classified as the second-order effects according to the nomenclature by Reiner and Abir [47].
The general format of the power threshold function applicable to advanced metals, having not only three threshold parameters a, b and c but also including two independent powers p and r in the yield/failure criterion for isotropic materials, can be furnished as follows, see Ganczarski and Skrzypek [15] and Skrzypek and Ganczarski [51]: It will be shown below that the majority of criteria met in the literature to predict onset of yield, failure or even phase transformation (Iyer [26], Pȩcherski et al. [42], Gao et al. [20], Iyer and Lissenden [27], Brünig et al. [4], Raniecki and Mróz [46]) can be captured as the specific cases of this general format (9). If a = 0 and r = 1 whereas p is arbitrary, the Raniecki and Mróz cylindrical surface is recovered from (9), or equivalently, if the original notation is saved: that represents a rounded yield surface with trigonal symmetry, see Fig. 1b. If, additionally, n = 2, the classical Drucker's [10] format is recovered, Assuming namely r = p in Eq. (9), the Iyer [26] yield/failure onset criterion is recovered, Equation (13) is successfully used to describe the Nickel-based alloy Inconel 718 at elevated temperature 650 • C by Iyer and Lissenden [27]. If r = p = 1, the narrower format used by Iyer and Lissenden [27], Pȩcherski et al. [42] is recovered, An equation of this type, which represents an asymmetric either paraboloidal or ellipsoidal surface, was calibrated by Pȩcherski et al. [42] for Inconel 718. Note that the Huber-von Mises f (J 2s ), the Drucker-Prager f (J 1σ , J 2s ) and the Drucker f (J 2s , J 3s ) yield functions can be obtained as special cases. Assuming another combination of powers p/r = 1/2 and r = 6, we arrive at the Gao et al. [20] yield function calibrated and verified for the 5083 aluminium alloy. The constant c 1 can be found by introducing the uniaxial condition onto equation (15), Another special case of Eq. (9), when p/r = 1/2 and r = 1, was considered by Brünig et al. [4], who confirmed its applicability for failure initiation in aluminium alloys and high-strength steels. An equation of this type represents an asymmetric cone, and it is capable of capturing yield onset in the high-strength 4310 and 4330 steels. If a = α, b = 0 and c = k, Eq. (17) reduces to the well-known Drucker-Prager equation An asymmetric yield function of pressure-sensitive materials, which cannot be obtained from Eq. (9), is due to Yoon et al. [60], Equation (19) is in fact an extension of Drucker's equation (12), by use of an additional term linear with respect to the first invariant J 1σ , according to experimental results by Spitzig et al. [53] as well as Spitzig and Richmond [54]. This format inherits after Drucker's trigonal symmetry that enables to describe the SD effect, but the presence of first invariant J 1σ leads to a conical surface, instead of a cylindrical in Drucker's case. Let us mention that from among three material constants a, b and c in (19), only two are independent since the third a has to capture the uniaxial tensile test, namely Additionally, the convexity of the proposed yield function is ]. This isotropic equation is also extended to material anisotropy in order to properly describe various metals of AA 2008-T4, high-purity α-titanium and the AZ31 magnesium alloy. In case if pressure sensitivity is ignored (b = 0 and a = k), the Cazacu and Barlat [6] format is recovered, 3 Influence of stress invariants on a description of pressure sensitivity, asymmetry and anisotropy: explicit or implicit approaches The "sharp" distinction between the ductile and brittle materials mentioned in Sect. 1 is often not exactly justified, especially when more advanced structural material behaviours are considered. Numerous experimental findings referred to novel materials suggest another classification with respect to distinct mechanical responses, among which the three are of particular importance: -hydrostatic pressure sensitivity, -tension versus compression asymmetry, -material anisotropy.
For the purpose of the present paper, the hydrostatic pressure dependence is considered as the most important among the three. In this sense, materials can be classified into two groups: hydrostatic pressure-dependent and hydrostatic pressure-independent materials, also called pressure-sensitive and pressure-insensitive ones. Traditionally, ductile materials (majority of metals) can be considered as hydrostatic pressure independent. On the other hand, brittle materials (rocks, ceramics, etc.) should be treated as hydrostatic pressure-dependent ones. Hydrostatic pressure dependence of isotropic or anisotropic limit criteria can be captured in the two different manners: 1b. the first common invariant plus the second common deviatoric invariant f [π : σ , s : : s] anisotropy.
2. indirect dependence by the second invariants with the first invariants ignored: 2b. the second common stress invariant However, there exists a broad class of engineering materials which do not exhibit any dependence on the hydrostatic pressure, either direct or indirect. This means that in case of the isotropic hydrostatic pressure-independent materials the corresponding limit surfaces have to include the second deviatoric invariant exclusively. In case of anisotropy limit, surfaces can include the second common deviatoric invariant and additionally the first common deviatoric invariant. In all cases considered, any equation of limit surface has to include the second stress or the second common invariants, which results from the quadratic form of energy representation. Exemplary equations of limit surfaces that found application in engineering materials are presented in Table 1 according to the aforementioned classification. The above Table shows a comparison between the pairs of appropriate isotropic and anisotropic criteria that correspond to: the direct dependence on hydrostatic pressure, e.g. Drucker-Prager's criterion versus Tsai-Wu criterion, the indirect dependence on hydrostatic pressure, e.g. Beltrami criterion versus von Mises criterion, and independence of the hydrostatic pressure, e.g. Huber-von Mises criterion versus Hill criterion. The oldest criterion based on total elastic energy, formulated by Beltrami in 1885, is invoked in this Table although it has no experimental evidence, cf.Życzkowski [61].
Tension/compression asymmetry, also called strength differential effect (see for instance Altenbach et al. [2]), is included in a natural way in limit criteria for anisotropic materials. In the case of limit criteria for isotropic materials, this effect manifests itself by the presence of first stress invariant J 1σ and/or the third stress invariant J 3s , as shown in Fig. 1. Note that in the case of the first stress invariant-dependent surface, compressive and tensile meridians are in identical distance from the centre of the limit curve, but the axis of the limit surface is shifted in Fig. 1a. In other case, when the limit surface is the third stress invariant-dependent function, compressive and tensile meridians are not in identical distance from the centre of the limit curve, but the axis of the limit surface remains at the position of the hydrostatic axis, see Fig. 1b. In our opinion, there is no sharp distinction between two terms appearing in the literature: the tension/compression asymmetry and the strength differential effect, such that both terms can be used equivalently. This might be confirmed by many authors: Raniecki and Mróz (SD effect due to J 3s ) [46], Cazacu et al. [7] (tension/compression asymmetry due to J 3s ), Khan et al. [30] (tension/compression asymmetry due to J 3s ) , Altenbach et al. [2] (tension/compression asymmetry = SD effect), to mention only some.
As it was aforementioned, the limit criteria have to include appropriate second invariants. However, limit surfaces based on the second invariants exclusively (stress invariants or common invariants) are capable to capture neither tension/compression asymmetry nor any shape change due to distortion. By contrast, the limit criteria based on the second and the third invariants (stress invariants or common invariants) are capable to capture both tension/compression asymmetry and distortion. Table 2 shows a comparison between the pairs of selected isotropic and anisotropic criteria that correspond to: the lack of tension/compression asymmetry and distortion, e.g. Huber-von Mises' criterion vs. Hill's criterion, tension/compression asymmetry with distortion ruled out, c.f. Drucker-Prager's criterion versus Tsai-Wu's criterion, and tension/compression asymmetry with distortion accounted for, c.f. Drucker's criterion versus Kowalsky's et al. criterion. To illustrate the classification described in Table 2, a comparison between asymmetry without distortion and asymmetry with distortion accounted for is presented in Fig. 2. In the case of isotropy Fig. 2a, the Drucker criterion is compared with the Tsai Drucker-Prager criterion. In the case of Drucker-Prager's criterion, based on the first and the second stress invariants, tension/compression asymmetry appears independently from shape distortion. By contrast, when Drucker's criterion is used, both effects are coupled by the third invariant, and hence they appear simultaneously.
In the case of anisotropy Fig. 2b, the Tsai-Wu criterion is compared with the Kowalsky et al. criterion. Tsai-Wu's criterion accounts for tension/compression asymmetry without distortion (only translation by the first common invariant is accounted for). By contrast, when the Kowalsky et al. sixth-order criterion is used, the tension/compression asymmetry and shape distortion are coupled in an anisotropic fashion by the third common invariant.
In general, material anisotropy can be captured by use of the two approaches. In the first mathematically consistent approach, here called the explicit anisotropy approach, the system of stress invariants J 1σ , J 2s and J 3s is substituted by the corresponding system of common invariants π : σ , s : : s and s : I¯I : s : s. In the other currently dynamically developed approach, here called the implicit anisotropy approach, developed in Barlat and Khan schools, either the second J 2s and the third J 3s stress invariants are substituted by the corresponding transformed deviatoric invariants J 0 2s , J 0 3s , or the stress deviator is transformed by use of the two independent fourth-rank transformation tensors Σ = C : s and Σ = C : s, and next they are inserted to one of the well-known isotropic criteria, either Drucker's one or Hosford's one, respectively. These linear transformations correspond to mapping of the deviatoric Cauchy stress tensor s to the other two deviatoric stresses Σ and Σ referring to the material anisotropy (orthotropy) frame. The implicit approach is able to capture the full material orthotropy (with distortion effect included) by use of two fourth-rank orthotropic transformation tensors C, C (containing 2 × 9 = 18 independent material constants), in contrast to the explicit common invariant-based approach which requires in the case of material orthotropy, a fourth-rank tensor and sixth-rank tensor I¯I (containing 9 + 56 = 65 material constants). Although the explicit approach is mathematically more rigorous than the implicit one, simultaneously it is much more cumbersome and compliant to misunderstandings. Both approaches, the explicit and the implicit, are alternatively used, but obviously they lead to different approximations. Comparison of the explicit and the implicit approaches to capture anisotropy is schematically presented for selected criteria in Table 3.
A major difficulty when describing the limit yield/failure is caused by the coupling between anisotropy and strong tension/compression asymmetry, as discussed by Khan et al. [30]. Such significant coupling can lead to a total distortion of the limit surface (possible lack of any axis of symmetry), as it is presented in Fig. 3, based on Luo et al. [35] experimental findings for AZ31B Mg alloy, and fitted by Plunkett et al. [44]. Another interesting "hybrid" concept is applied by Yoon et al. [60]. It is a combination of both the explicit approach in the case of first common invariant and simultaneously the implicit approach with regard to second and third transformed invariants. In Yoon et al. [60], the orthotropic yield criterion and the isotropic criterion being its original pattern are compared in Table 4.
This highly extended yield criterion is capable of capturing all three features: anisotropy, tension/compression asymmetry and hydrostatic pressure sensitivity of various metals like: AA 2008-T4, highpurity α-titanium, and AZ31 magnesium alloy. Excellent fitting of the proposed yield criterion and experimental data for AZ31 alloy is shown in Fig. 3. This sufficiently general form can unconditionally be recommended as very effective and especially addressed to model the totally distorted response of cold rolled metals.
However, some limitations for the implicit approach can be pointed out. In the case of the explicit formulations, the conditions that guarantee convexity of the anisotropic limit surface are known, see Ottosen and Ristinmaa [40], Ganczarski and Skrzypek [19]. In the other case of an implicit formulation, general conditions for convexity are very difficult and cumbersome to reach, Cazacu et al. [6,7]. This is due to a substitution of isotropic stress invariants by the linearly transformed stress, Plunkett et al. [44], Yoon et al. [60].
Additionally, the use of doubled linear stress transformation has no clear physical sense, although this guarantees perfect fitting to experimental data. Finally, although both curves in Fig. 3 describe the same material in the σ x x , σ yy plane, corresponding 3D limit surfaces belong to different classes: cylindrical surface for pressure sensitivity vs. conical surface for pressure sensitivity.
Reassuming the above broad discussion concerning three essential aspects: hydrostatic pressure sensitivity, strength differential effect and material anisotropy, emphasizes that there exists strong coupling between them. In general, at least two ways to capture these coupled features: the conventional common invariants formulation or the linear transformed stresses to account for anisotropy, can be distinguished. Coupling between all three mechanical responses manifests strongly when implicit formulation is used. This is due to the non-separateness of the above effects involved by the transformation tensors applied to all invariants. By contrast, when the explicit formulation is applied, it is possible to point out that the first common invariant is directly responsible for hydrostatic pressure sensitivity and strength differential effects. However, to account for the distortion Table 3 Explicit or implicit anisotropy of limit surfaces Explicit Implicit

Iyer and
Hosford [25] Lissenden [27] Isotropy s : I Plunkett et al. [44] Anisotropy s  3 Comparison of the different approaches to modelling limit curves for AZ31B Mg alloy: a hydrostatic pressure-insensitive criterion by Plunkett et al. [44], fitting of Luo et al. [35] experimental data, and b hydrostatic pressure-sensitive criterion by Yoon et al. [60], (unit MPa) effect it is necessary to apply at least the third common invariant. In what follows, the explicit approach will consistently be applied.

Von Mises-Tsai-Wu-type yield/failure criteria
In what follows, the narrower format of Eq. (4) with the third common invariant σ : I¯I : σ : σ neglected is used, π : σ + σ : : Removal of the third common invariant excludes distortion of the limit surface. In other words, limit curves are ellipses instead of ear-like shapes, as shown in Fig. 3. The structural tensors of the second π and fourth orders appearing in Eq. (26) stand for two independent yield/failure anisotropy tensors, the identification of which has to be performed on the basis of respective yield/failure tests in an analogous way as that discussed in [16]. However, in the present case two anisotropy tensors have to be calibrated, and hence the appropriate number of tests increases, such that the difference between the tension and the compression uniaxial tests can be captured. Substituting for convenience Voigt's vector-matrix notation, both above tensors can be represented in the vector-matrix forms [π ] = ⎡ ⎣ π 11 π 12 π 13 π 22 π 23 π 33 ⎤ ⎦ or {π} T = π 11 , π 22 , π 33 , π 23 , π 13 , π 12 T (28) and where yield/failure loci are determined by two yield/failure characteristic matrices [π] of the dimension (3×3) and [ ] of the dimension (6 × 6). Hence, in the considered case of general anisotropy, the number of moduli defining yield/failure initiation is equal to 27 = 6 + 21.
The condition of yield/failure initiation in anisotropic materials (26) takes in Voigt's notation the equivalent format being an extension of the anisotropic von Mises' criterion of plastic materials, however, enriched by the linear term.
Note that this form is strictly pressure insensitive only in the quadratic terms, but it is pressure sensitive as far as the linear terms are concerned. The fully deviatoric format of Eq. (32) by an additional constraint applied for the linear term π 11 + π 22 + π 33 = 0 was discussed by Szczepiński [55], whereas an experimental verification for low carbon steel 18G2A is due to Kowalewski andŚliwowski [31].
Limiting further considerations to orthotropic materials, both characteristic matrices [ ort π] (27) and [ ort ] (29) take the following forms valid for principal directions of orthotropy: The second-rank matrix [ ort π] is of the dimension 3 × 3, whereas the fourth-rank matrix [ ort ] is of dimension 6 × 6. The matrix [ ort π ] has diagonal form, and the matrix [ ort ] is of the identical symmetry as the von Mises plastic orthotropy matrix. Both matrices (36) are defined by 12 = 3 + 9 modules. The condition of yield/failure initiation for orthotropic materials takes a form typical for the rotationally symmetric group being an extension of the von Mises orthotropic yield condition which is pressure insensitive, for pressuresensitive materials. The von Mises orthotropic yield/failure initiation criterion (37) can be written down in the following extended format: Note that the above equation represents the fully tensorial form of the orthotropic yield/failure criterion, contrary to the deviatoric form which is characteristic for the Hill yield criterion. This means that 12 = 3 + 9 material moduli defining the yield/failure material characteristic tensors ort π and ort are required for its identification.
Consider now a reduction of the orthotropic criterion (37) to a narrower format known in the literature as the Tsai-Wu orthotropic criterion of failure. The Tsai-Wu criterion is characterized simultaneously by the strength differential effect and pressure insensitivity of [ ort ] in Eq. (36) This leads to the following narrower representation of both characteristic matrices: The form of Eq. (39) and its representation (40) reflects a "hybrid notation" in the following sense: the first term represents the linear common invariant of the stress tensor σ and the structural tensor π TW (analogy to the pressure sensitivity in case of isotropic material), whereas the second term represents the quadratic common invariant of the stress deviator s and the structural tensor TW (defining shape and orientation of the surface in the stress space). The criterion (39) takes therefore the explicit format of the nine-parameter Tsai-Wu's criterion [58], As a matter of fact, any addition of a hydrostatic pressure to all normal stresses σ x → σ x ± σ h does not change the magnitude of quadratic terms in condition (41), but simultaneously causes the linear terms still to depend on σ h . Hence, finally the Tsai-Wu criterion in the format given by (41) remains the pressure-sensitive one by the linear terms. This condition can be considered as a special case of the n-type yield functions presented by Pariseau [41], see Chen and Han [8]. The Pariseau condition predicts a generalized paraboloidal increase of strength with hydrostatic pressure. However, the conventional quadratic format of the Pariseau criterion, commonly named the Tsai-Wu criterion [58], was identified and implemented to ice crushing failure by Ralston [45].

Transversely isotropic Tsai-Wu-type format: tetragonal versus hexagonal symmetry criteria: convexity analysis
Similarly to [16], a further reduction of the nine-parameter yield/failure orthotropic Tsai-Wu's criterion (41) to the narrower format of transverse isotropy requires a precise distinction between the tetragonal and hexagonal symmetry classes. Assuming, after Chen and Han [8], the plane of transverse isotropy x y, the fourth-rank orthotropy matrix [ TW ] (40) reduces to the transversely isotropic format [ tris TW ], analogously to the transversely isotropic Hill criterion possessing only four independent material constants, whereas the second-rank transversely isotropic matrix [ tris π TW ] reduces to a form possessing only two independent material constants. Namely, assuming Π 23 = Π 13 , Π 44 = Π 55 and π 11 = π 22 in (40), we arrive at the six-parameter form of the transversely isotropic yield/failure matrices of tetragonal symmetry In case when the non-abbreviated notation is used, the six-parameter transversely isotropic Tsai-Wu yield/failure criterion of tetragonal symmetry takes the following format: Equation (43) is expressed in terms of six material anisotropy modules: Π 12 , Π 13 , Π 44 , Π 66 , π 11 and π 33 , which will be determined in terms of six independent material coefficients, referring to appropriate tensile and compressive strengths k tx , k cx , k tz , k cz and shear strengths k zx , k xy . In order to calibrate them, the following tests have to be performed: which lead to the system of equations: The solution of Eqs. (45), with respect to Π 13 , Π 12 , Π 44 , Π 66 , π 11 and π 33 , yields The final form of the tetragonal transversely isotropic Tsai-Wu criterion in terms of k tx , k cx , k tz , k cz , k zx and k xy is Except the above tetragonal format of the transversely isotropic Tsai-Wu criterion Eq. (47), there exists also its hexagonal format, in which the number of essentially independent moduli is equal to five. In case if the narrower hexagonal symmetry sub-class is assumed, the sixth diagonal modulus Π 66 has to satisfy the relationship (cf. Chen and Han [8], Ganczarski and Skrzypek [14]) The constraint (48) satisfies the reducibility of the criterion (47) to the format invariant with respect to equivalent simple shear stress states: τ xy = σ and σ x = σ, σ y = −σ , which must hold in the transverse isotropy x y plane. The sixth material modulus Π 66 is equal to The final format of the hexagonal transversely isotropic Tsai-Wu criterion in terms of k tx , k cx , k tz , k cz and k zx is furnished as Both Tsai-Wu limit equations: the tetragonal format (47) and the hexagonal format (50) may suffer from a possible convexity loss, analogously as in the case of corresponding transversely isotropic Hill's criteria, as discussed in detail in [16]. However, in the previous case the strength differential effect was not taken into account, such that the known inequality bounds by Ottosen and Ristinmaa [40] could be applicable, For the present convexity analysis, the extended inequality bounds the range of applicability of the transversely isotropic Tsai-Wu criteria (47) and (50), where the strength differential effect is included. It should be noticed that by satisfying the condition (52) convexity of both criteria (47) and (50) is guaranteed. In other words, criterion (52) means that the material coefficients in the x y transverse isotropy plane, preceding both terms σ x σ y and the τ 2 xy in (50), cannot change signs.  Fig. 4 Degeneration of the Tsai-Wu limit surface with the magnitude of the generalized Hosford and Backhofen parameter R: a transverse isotropy plane, b orthotropy plane, after Ganczarski and Adamski [17], Skrzypek and Ganczarski [51] A general proof of convexity of the Tsai-Wu criterion (41), as well as its particular case of transverse isotropy (43), is presented in Appendix A.
Substitution of the dimensionless parameter being an extension of the Hosford and Backhofen parameter leads to the simplified format of (52) If the above inequality (54) does not hold, elliptic cross sections of the limit surfaces degenerate to two hyperbolic branches such that loss of convexity occurs.
To illustrate what happens if criterion (54) is violated, the yield/failure curves in two planes: the transverse isotropy plane (σ x , σ y ) (55) and the orthotropy plane (σ x , σ z ) for various R-values, are sketched in Figs. 4a and 4b, respectively. Note that in the case considered, when shear terms which distinguish both formulas, tetragonal (47) and hexagonal (50), do not appear, Eqs. (55) and (56) are identical for both formulations. It is observed that, when R is ranging from R = 3 to R = −0.5, the limit curves degenerate from closed ellipses to two parallel lines, whereas for R < −0.5, the inadmissible concave hyperbolas appear.

New formulation of unconditionally convex von Mises-Tsai-Wu-type criterion
Both previously discussed criteria: the tetragonal (47) and the hexagonal (50) suffer from possible convexity loss, moreover criterion (50) exhibits an intrinsic contradiction since it promises transverse isotropy of hexagonal type, whereas the ratio between the ellipse semi-axes in this plane is not equal to the corresponding ratio of the isotropic Huber-von Mises ellipse, analogously as it was proposed in [16]. In order to avoid both inconveniences, in what follows the other hexagonal transversely isotropic von Mises-Tsai-Wu yield/failure criterion, see Ganczarski and Adamski [17], Skrzypek and Ganczarski [51], is proposed. This is unconditionally convex and simultaneously preserves the Huber-von Mises ratio of ellipses' semi-axes. In order to do this, consider the more general transversely isotropic von Mises-Tsai-Wu criterion of the format Π 11 σ 2 x + σ 2 y + Π 33 σ 2 z + 2Π 12 σ x σ y + 2Π 13 σ x + σ y σ z + Π 44 τ 2 yz + τ 2 zx + Π 66 τ 2 xy + π 11 σ x + σ y + π 33 σ z = 1.
Equation (57) contains 8 = 6 + 2 modules, and it is a straightforward simplification of the orthotropic von Mises-Tsai-Wu criterion (38) by assuming Π 11 = Π 22 , Π 23 = Π 31 , Π 44 = Π 55 and π 11 = π 22 . For calibration of (57), the following tests can be performed: four uniaxial tension/compression tests and one out-of-plane shear test, Additionally, three lacking conditions: one in-plane shear condition and two biaxial conditions that allow to capture magnitudes of Π 66 , Π 12 and Π 13 , are assumed. Symbols Δ 1 and Δ 2 , used for brevity, denote: The solution of Eqs. (58)(59)(60) with respect to eight material moduli Π 11 , Π 12 , Π 13 , Π 33 , Π 44 , Π 66 , π 11 and π 33 yields which finally leads to the new hexagonal transversely isotropic von Mises-Tsai-Wu failure criterion also in terms of 5 independent material constants k tx , k cx , k tz , k cz and k zx , but different from hexagonal format of the Tsai-Wu criterion (50), Criterion (63) ensures unconditional convexity, regardless of the magnitude of orthotropy ratio, and required coincidence of the semi-axes ratio, in the isotropy plane with that expected from the isotropic Huber-von Mises criterion, see  (58) and (60) applied, are compared for columnar ice, the experimental data of which were established by Ralston [45] in Table 5, in the plane of transverse isotropy (σ x , σ y ) and the shear plane (σ x , τ xy ), see Fig. 6, and in the plane of orthotropy (σ x , σ z ), see Fig. 7. Corresponding cross sections of the limit surface are ellipses that exhibit strong oblateness, the centres of which are shifted outside the origin of the coordinate system towards the third quarter. In the case of cross section by the plane of transverse isotropy, in Fig. 6, the symmetry axis is inclined at 45 • . In other words, it overlaps the projection of the hydrostatic axis at the transverse isotropy plane (σ x , σ y ), contrary to the cross section by the plane of orthotropy, in Fig. 7, where the main semi-axis of the ellipse is inclined by 71.1 • . It has to be emphasized that, in the case of columnar ice, the compressive strength along the orthotropy axis k cz is more than 10 times larger than tensile strength k tz , whereas the analogous ratio k cx /k tx is approximately equal to 7 in the case of the transverse isotropy plane. Both the hexagonal transversely isotropic Tsai-Wu failure criterion Eq. (50) and the new hexagonal transversely isotropic von Mises-Tsai-Wu failure criterion Eq. (63) contain the same number of 5 independent strengths k tx , k cx , k tz , k cz and k zx ; however, only the criterion (63) is unconditionally convex.

Remarks on the implicit formulation of pressure-sensitive anisotropic failure criteria: Khan's concept
In what follows, selected examples of implementation of the implicit approach to the broader class accounting for anisotropy, tension/compression asymmetry, and pressure sensitivity are thoroughly considered. In some cases of anisotropic alloys exhibiting tension/compression asymmetry, it is convenient to consider a scalar function of selected (mixed) stress invariants and common invariants (cf. Khan and Liu [29], etc.), f tr(σ ), 1 2 tr(s · s), 1 3 tr(s · s · s); . . . , σ : : σ , . . . .
The particular format of (64) can be recognized as a combination of (6) and (25). Khan and Liu [29] applied the following extension of the nine-parameter orthotropic von Mises criterion to describe the ductile failure of the Ti-6Al-4V alloy accounting for: hydrostatic pressure sensitivity, anisotropy and significant tension/compression asymmetry effect: Both the hydrostatic pressure dependence I 1 and the tension/compression asymmetry J 3 are included in an implicit fashion as arguments of two exponential functions, appearing as multipliers at the right-and the left-hand sides of the orthotropic von Mises' equation. According to the authors' interpretation, the main advantage of such formulation is that the anisotropy and tension/compression asymmetry are uncoupled into separate multiplicative terms, which allow the anisotropic parameters and tension/compression asymmetry coefficient to be determined independently. The following definitions hold: F, G, H, L , M, N , P, Q and R are anisotropic parameters, C is the tension/compression asymmetry coefficient, ζ denotes the Lode parameter ζ = cos 3θ = 27 2 J 3 ( √ 3J 2 ) 3 , where θ is the Lode angle, I 1 is the first stress invariant, whereas J 2 and J 3 are the second and the third invariants of the deviatoric stress tensor. Although the general form of limit criterion (65), accounts for all three features: anisotropy, tension/compression asymmetry and hydrostatic pressure dependence, in fact its calibration performed by the authors leads to the form capturing only the tension/compression asymmetry and hydrostatic pressure dependence. As a consequence, the limit curve of Al2024-T351 alloy exhibits only one axis of symmetry, which means that this corresponds to the case of a partly distorted limit surface. By the use of the above formula, the authors succeeded with fitting experimental data in rolling direction (RD), transverse to rolling direction (TD), and the thickness direction (ND).
However, hydrostatic pressure dependence introduced by the use of the right-hand-side exponential function leads to loss of convexity of the failure surface along the meridional direction as it was shown by Khan and Liu [29] in Fig. 10b in Appendix B. This property was broadly discussed, where an analogy between anisotropic Khan-Liu's and isotropic Burzyński's surfaces was presented. An essential difference between these two cases is pointed out, because the Burzyński surface exhibits only conditional meridional convexity loss, whereas the Khan-Liu surface exhibits unconditional concavity built-in. The convexity loss discussed in this case is significant only from the theoretical point of view because in such a case Drucker's postulate is violated. However, for the data cited by the authors the concave meridian effect is very small such that it can probably be ignored from the engineering point of view for the considered material data. Nevertheless, in spite of possible convexity loss along the meridian, none convexity loss along circumference is observed although there exists a second exponential function dependent on J 2 and J 3 , being a multiplier of the Hill form on the left-hand side of Eq. (65).
In another paper by Khan et al. [30], the direct hydrostatic pressure dependence (by I 1 ) is dropped; however, both significant anisotropy (fully anisotropic calibration of all material constants F, G, . . . , R) and tension/ Although the general form of the limit criterion (66) accounts for nine independent anisotropy parameters, in the example considered by the authors in [30], due to calibration the material constant G is determined from the equi-biaxial compression test; so, it depends on three compression limits like in the case of Hill's criterion. Under the assumption of plane stress state, it reduces to the four-parameter orthotropic Hill's condition. Fitting of experimental data for Ti-6Al-4V alloy at different strain rates and temperatures shows excellent coincidence between the experimental findings and simulation. By contrast to the previous formulation (65), the symmetry of the limit curve is completely lost (no axis of symmetry exists), as shown in Fig. 8.

Conclusions
The present contribution is a straightforward continuation of the earlier work by Ganczarski and Skrzypek [16]. This is an essential extension for pressure-sensitive materials, instead of the previously discussed pressureinsensitive materials. Detailed discussion on a possible convexity loss of the conventional Tsai-Wu criterion is done. The Tsai-Wu-and Pariseau-type criteria describe materials in which failure limits in tension and compression may be essentially different. Hence, these criteria are strongly exposed to a possible convexity loss, see Eq. (50). To avoid the aforementioned deficiency of convexity lack, the new von Mises-Tsai-Wu failure criterion, which is free from the possibility of inadmissible degeneration of the single convex, simply connected elliptical surface into two concave hyperbolic surfaces as shown in Eq. (63), is proposed. This was successfully done by relaxing the constraint (48) for Π 66 , which is no longer dependent on Π 13 and Π 12 .
The paper is focused on the three aspects: pressure sensitivity, strength differential effect, and anisotropy. The authors are convinced of the conventional approach, which is based on rigorous application of common invariants formalism, see Goldenblat and Kopnov [21], Sayir [50], and others.
Due to dynamic growth in the field of material fabrication and design, including metallic ones that exhibit all three properties, the parallel approach based on the transformed stress applied to isotropic limit criteria is developed, see Barlat et al. [6,7,44], Khan et al. [30], and others. Also, other mixed-type criteria, which combine some elements from the theory of common invariants and the approach based on transformed stress, are developed in parallel, see Yoon et al. [60]. Having this in mind, in this paper attention is paid to compare the traditional rigorous approach versus the transformed stress approach, see Appendix C.
General conclusions can be drawn as follows: -newly formulated constitutive laws must follow fundamental laws of symmetry and convexity, -clear physical bases are necessary to include, -even impressive fitting of a proposed single curve to experimental data may not lead to a correct description of the limit surface as a whole, -the particular case of transverse isotropy should be considered carefully, because of its important place in the field of composites, and requires the precise distinction between symmetry groups and classes, the reducibility to well-established isotropic formulations, -continuous symmetry (infinite number of symmetry axes) should obey in the transverse isotropy plane for the criterion proposed, preserving the ratio of semi-axes identical to the isotropic Huber-von Mises ellipse. As a matter of fact, this paper addresses the initial limit surfaces only, but not a loading process on which changes in the subsequent limit surfaces occur. It means that the convexity requirement (Sylvester or Drucker) is guaranteed strictly at the beginning of the hardening process that produces microstructural changes. Hence, the convexity requirement for the initial limit surface may but not has to guarantee convexity of subsequent surfaces. On the other hand, a concavity of the initial surface must result in a lack of convexity of subsequent surfaces. where σ stands for stress tensor, whereas π TW and TW denote structural tensors of material anisotropy of second and fourth rank, respectively. The condition of convexity for the yield/failure function (A.1) is that the Hessian matrix ought to be positive semi-definite, To save the generality of convexity proof, in what follows, the arbitrary stress components are considered, instead of the conventionally used principal stress components, see Cazacu et al. [7]. The format of the Tsai-Wu yield/failure function (A.1), including linear and quadratic terms with respect to the stress tensor, inserted to (A.2), shows that only the fourth-rank structural tensor of material anisotropy decides convexity, Note that the third (cubic) invariant in (A.5) is equal to zero, which confirms hydrostatic pressure independence of the Hessian and leads to the following spectral decomposition (eigenvalues): It is visible that only one of three eigenvalues (A.6) may be negative, and hence the fourth inequality completing (A.4) is as follows: The conditions of convexity (A.4) and (A.7) may be expressed in terms of tension, compression and shear strength, namely performing 9 known tests magnitudes of moduli of material orthotropy Π i j can be established as follows: which inserted to (A.4) and (A.7) leads to the inequalities 2 (k tx k cx )(k ty k cy ) which turn out to be generalizations of Ottosen and Ristinmaa's [40] bounds. In the case of transverse isotropy, the symmetry conditions k tx = k ty and k cx = k cy reduce Eq. (A.9.1) to and simultaneously Π 66 = −2(Π 13 + 2Π 12 ) inserted to (A.8.6) leads to which is obviously the sub-case of (A.9.1). Finally, substitution of the generalized Hosford and Backhofen parameter

Appendix B: Comparison of possible meridional convexity loss of limit surfaces by Khan-Liu and Burzyński
In Sect. 6, the unconditionally convex von Mises-Tsai-Wu criterion has been formulated and discussed, in contrast to Sect. 5, where the Tsai-Wu criterion exhibiting possible convexity loss in hoop direction has been discussed in detail. The majority of researchers notices the case of inadmissible hoop convexity loss, in the light of Drucker's postulate sense, for instance Cazacu and Barlat [6], Ottosen and Ristinmaa [40], Cazacu et al. [7], Ganczarski and Lenczowski [13], Ganczarski and Skrzypek [19], Skrzypek and Ganczarski [51], to mention only some of them. However, very few concepts deal with the other case of meridional convexity loss.
In what follows, the above feature incorporated to both the anisotropic Khan-Liu concept and chronologically earlier isotropic Burzyński criterion is discussed. Either Khan-Liu or Burzyński criteria are obviously pressuresensitive ones. It will be shown that Burzyński's criterion, which is usually convex, may loose convexity in particular cases, whereas Khan-Liu's criterion possesses a concavity built-in. The three-parameter isotropic criterion, rotationally symmetric yield/failure surface, originally introduced by Burzyński [5], is where A, B, and C are the material constants. These constants are determined based on three tests: the uniaxial tension (k t ), the uniaxial compression (k c ), and the simple shear (k s ). These calibrations lead to the general format of the three-parameter Burzyński's criterion, Such formulation presumes not only tension/compression asymmetry k t = k c , but also the third shear limit point k s is considered as independent. Equation (B.2) represents different types of Burzyński's rotationally symmetric surface depending on mutual relationships between k t , k c and k s . In case if the shear yield/failure strength k s is larger than √ k t k c /3, Eq. (B.2) represents a rotationally symmetric ellipsoid, if the shear yield strength is equal to √ k t k c /3 it represents a rotationally symmetric paraboloid, whereas if the shear yield strength is less than √ k t k c /3 it represents a twofold rotationally symmetric hyperboloid. If k s reaches its lower admissible bound 2k t k c √ 3(k t +k c ) , the Burzyński hyperboloid transforms to the Drucker-Prager's cone. Shear yield strengths less than 2k t k c √ 3(k t +k c ) are inadmissible in the sense of Drucker's postulate, since in such case Eq. (B.2) represents a onefold concave hyperboloid (Fig. 9). In the case of a twofold hyperboloid, only one fold that includes stress origin has physical sense.
The two-parameter paraboloidal approximation of Burzyński's surface (k s = √ k t k c /3) can be written as follows: It is experimentally verified for metallic alloys, see for instance material constants for Inconel 718 cited by Pȩcherski et al. [42], Note that the yield/failure strength satisfies condition k s = √ k t k c /3. In the other particular case, if the two-parameter conical approximation of Burzyński's surface is furnished, The above condition can be reduced to the Drucker-Prager's condition commonly met in the literature. Note, however, that in the light of the above discussion the Drucker-Prager's condition can be considered as the limit case for the applicability of Burzyński's criterion. Below this limit, when k s < 2k t k c √ 3(k t +k c ) , the conical surface deforms into the concave onefold hyperboloid, which is inadmissible following Drucker's stability postulate.   [9], Mohr [38] |τ | = c − σ tan φ Format, Chen and Han [8] + 1 3 J 2s cos θ + π 3 sin φ − c cos φ = 0 C6 Burzyński [5] A3J 2s + B J 1σ 3 2 + C J 1σ 3 = 1