Abstract
Secondorder work analyses, based on elastoplastic models, have been frequently carried out leading to the result that failure may occur before the limit yield condition is encountered. In this article, secondorder work investigations are carried out with barodesy regarding standard element tests and finite element applications. In barodesy, it was shown—like in hypoplasticity and elastoplasticity—that secondorder work may vanish at stress states inside the critical limit surface. For boundary value problems, an endtoend shear band of vanishing secondorder work marks situations, where failure is imminent.
Keywords
Barodesy Constitutive model Finite element simulations Secondorder work1 Introduction
Investigations, based on elastoplastic models, have been carried out by several authors [2, 3, 4, 10, 17, 25, 35] leading to the result that failure may occur before the limit yield condition is encountered. Secondorder work investigations with hypoplasticity showed similar results [2, 7, 14, 28], among other things that for loose soil secondorder work vanishes at stress states inside the critical limit surface.
The literature on uniqueness, stability, bifurcation and failure is vast. Stability refers to systems, as characterized by their boundary conditions, and not only to materials. In particular, the nature of tractions on the boundary plays an important role, e.g. the question whether they are dead or follower loads or not. In this article, we consider the secondorder work expressed as \({\hbox {tr}}({\mathring{{\mathbf {T}}}}{\mathbf {D}})\). For symbols and notation, see Sect. 2. Many relevant citations can be found in Hill [9]. As they refer mainly to elastic solids, we cite some publications from soil mechanics.
Lade [15], based on experimental results, concludes that the violation of the stability criteria of Hill, \({\hbox {tr}}({\mathring{{\mathbf {T}}}}{\mathbf {D}})\ge 0\), and Drucker, \({\hbox {tr}}({\mathring{{\mathbf {T}}}}{\mathbf {D}}^{{\mathrm{pl}}})\ge 0\), does not necessarily evoke an observable collapse of the sample.
Negative secondorder work denotes softening, i.e. the tangential stiffness has at least one negative eigenvalue. For some static boundary conditions of dead loads, this implies increasing in kinetic energy and thus collapse. In fact, Nicot et al. [26] correlate vanishing secondorder work with increase of kinetic energy and corresponding failure.
2 Symbols and notation
We use the symbolic notation for Cauchy effective stress \({\mathbf {T}}\) and stretching \({\mathbf {D}}\). In the figures, the more familiar symbol \(\sigma _i\) instead of \(T_i\) is used for the principal stresses. Normal stresses are defined negative for compression. Tensors are written in bold capital letters (e.g. \({\mathbf {X}}\)). \({\mathbf {X}}:=\sqrt{{\hbox {tr}}{\mathbf {X}}^2}\) is the Euclidean norm of \({\mathbf {X}}\), \({\hbox {tr}}{\mathbf {X}}\) is the sum of the diagonal components of \({\mathbf {X}}\). The superscript 0 marks a normalized tensor, i.e. \({\mathbf {X}}^0 = {\mathbf {X}}/{\mathbf {X}}\). \({\mathbf {1}}\) denotes the secondorder unit tensor. Stresses are considered as effective ones, the normally used dash is omitted. The stretching tensor \({\mathbf {D}}\) is the symmetric part of the velocity gradient. Stretching \({\mathbf {D}}\) is only approximately equivalent to the strain rate \({\dot{\varvec{\varepsilon }}}\). For rectilinear extensions however, \({\mathbf {D}}\) equals \({\dot{\varvec{\varepsilon }}}\), considering the logarithmic strain \(\varvec{\varepsilon }\). \(p:=\frac{1}{3}{\hbox {tr}}{\mathbf {T}}\) is the mean effective stress, \(\varepsilon _{\text {vol}}={\hbox {tr}}\varvec{\varepsilon }\) is the volumetric strain. For compressive strain, \(\varepsilon _i\) is defined negative.
The deviatoric stress is written as \(q=(\sigma _1\sigma _3)\) and the deviatoric strain reads \(\varepsilon _q=2/3\cdot (\varepsilon _1\varepsilon _3)\) for axisymmetric conditions.
3 Barodesy
As in Sect. 4 only rectilinear extensions are examined, the corotational, objective stress rate \(\mathring{{\mathbf {T}}}\) coincides with \(\dot{{\mathbf {T}}}\). This follows from \({\mathring{{\mathbf {T}}}}={\dot{{\mathbf {T}}}}+{\mathbf {T}}{\mathbf {W}}{\mathbf {W}}{\mathbf {T}}\) and \({\mathbf {W}}={\mathbf {0}}\). In general cases (e.g. the finite element applications in Sect. 5), we do consider the Zaremba–Jaumann rate \({\mathring{{\mathbf {T}}}}\).
Several rectilinear deformations represented by the principal stresses \(T_1\), \(T_2\), \(T_3\) will be numerically examined as to whether there can be found \({\mathbf {D}}^0\)tensors such that tr\(({\dot{{\mathbf {T}}}}{\mathbf {D}}^0)= 0\). The boundary of the region in stress space with tr\(({\dot{{\mathbf {T}}}}{\mathbf {D}}^0)> 0\) is the surface of vanishing secondorder work.
4 Element tests
We investigate the secondorder work for specific loading paths in standard element tests (Sects. 4.1–4.2) and give a more general perspective in Sect. 4.3.
4.1 Undrained triaxial test
In Appendix 2, we add the drained triaxial test as an illustrative example to investigate secondorder work in barodesy.
4.2 Nonconventional drained triaxial tests
We consider drained triaxial tests with reduction of p at \(q=\) const. For normally consolidated Weald clay (for parameters see Table 1) we set \(p_{\text {ini}}=50\) kPa, \(p_{\text {ini}}=100\) kPa and \(p_{\text {ini}}=200\) kPa. The tests start as conventional drained triaxial tests and at \(\sigma _1=1/K_0\cdot \sigma _2\) the mean effective stress p is decreased by increasing the pore pressure, cf. similar experiments by Lade [15] and simulations by Wan and Pinheiro [34]. A reduction in the mean stress is obtained e.g. in the case of an excavation [8]. For the tests in Fig. 2a, the deviatoric stress remains constant (\(q=75\) kPa for test A, \(q=150\) kPa for test B, \(q=300\) kPa for test C), hence \({\dot{q}}=0\). The secondorder work according to equation 37 simplifies thus to \({\hbox {tr}}({\dot{{\mathbf {T}}}}{\mathbf {D}})={\dot{p}} \cdot {\dot{\varepsilon }}_{\text {vol}}\). With decreasing p, i.e. \({\dot{p}} \ne 0\), the secondorder work vanishes for this specific loading path at \({\dot{\varepsilon }}_{\text {vol}}=0\). Simulations with barodesy show that in the nonconventional drained triaxial tests of Fig. 2, the secondorder work vanishes inside the critical limit surface.
4.3 Investigations in the deviatoric plane
The following analysis has been carried out numerically. We consider the deviatoric plane \({\hbox {tr}}{\mathbf {T}}=500\) kPa \(=\) const. in the principal stress space spanned by \(T_1, T_2, T_3\) and we search for the boundary of the region where tr\(({\dot{{\mathbf {T}}}}{\mathbf {D}})>0\). We examine stress rays starting from the hydrostatic axis \(T_1=T_2=T_3\). On each ray, we step forward with small increments of deviatoric stress. At each step, we check whether there are \({\mathbf {D}}\) tensors such that the condition tr\(({\dot{{\mathbf {T}}}}{\mathbf {D}})=0\) is fulfilled.

For normally consolidated soil^{4} (OCR \(=1\) and \(e>e_{\text {c}}\)) this cone lies inside the cone of critical stress states (Eq. 25).

For \(e=e_{\text {c}}\) (\(p_{\text {e}}/p=2\) respectively), the cone of critical states and the cone of \({\hbox {tr}}{({\dot{{\mathbf {T}}}} {\mathbf {D}})}=0\) differ only slightly: However, the cone of vanishing secondorder work lies inside the cone of critical stress states.

For highly overconsolidated soil \(e<e_{\mathrm{c}}\) (OCR = 6), the cone with \({\hbox {tr}}{({\dot{{\mathbf {T}}}} {\mathbf {D}})}=0\) lies outside the cone of critical stress states.
For the nonconventional triaxial tests in Fig. 2, \(\beta\) is approximately \(86.4^\circ\) and the mobilized friction angle \(\varphi _{W_2}\) is approximately \(22.340^\circ\), which is also lower than the critical friction angle of Weald clay with \(24^\circ\). A variation of \({\mathbf {D}}\) according to Eqs. 21–23 in order to find vanishing values of \(W_2\) results in a slightly lower mobilized friction angle of \(\varphi _{W_2}=22.336^\circ\).

In Fig. 4 a 3D representation of surfaces formed by \({\hbox {tr}}{({\dot{{\mathbf {T}}}} {\mathbf {D}})}=0\) for three different overconsolidation ratios (OCR \(=1\), OCR \(=2\) and OCR \(=6\) according to Fig. 3) is shown. The cross section of the critical stress surface of barodesy (Eq. 25) with the deviatoric plane \({\hbox {tr}}{\mathbf {T}}=500\) kPa is added.

Figure 5a shows the mobilized friction angles \(\varphi _{W_2}\) (obtained with \(\sin \varphi _{\mathrm{m}}=\frac{T_{\text {min}}T_{\text {max}}}{T_{\text {min}}+T_{\text {max}}}\)) along the \({\hbox {tr}}({\dot{{\mathbf {T}}}}{\mathbf {D}})=0\) locus versus \(\alpha _\sigma\). For normally consolidated samples (OCR \(= 1\)), the minimum mobilized friction angle is \(\varphi _{W_2}\approx 18^\circ\), which is only \(3^\circ\) higher than the mobilized friction angle under oedometric conditions estimated with Jáky’s relation.^{5} Similar results have been obtained with hypoplasticity [7]. For the OCR \(=2\), the cone of vanishing secondorder work lies slightly inside the cone of critical states, cf. Fig. 5a. For highly overconsolidated soil, \(\varphi _{W_2}\) is higher than \(\varphi _{W_2}\) of the critical stress surface, cf. Fig. 5a.

Furthermore for OCR \(=1\) the angle \(\beta\) between normalized stress \({\mathbf {T}}^0\) and stretching \({\mathbf {D}}^0\) according to Eq. 27 is \(63^\circ<\beta <69^\circ\), cf. Fig. 5b, the lower the void ratio (the higher the OCR), the higher the angle \(\beta\). For highly overconsolidated soil, the angle \(\beta\) (\(77^\circ<\beta <82^\circ\)) in Fig. 5b is higher than for slightly overconsolidated or normally consolidated soil.

Figure 5c shows the dilatancy \(\delta ={\hbox {tr}}{\mathbf {D}}^0\) in dependence of the Lode angle \(\alpha _\sigma\). For normally consolidated clay (OCR \(=1\)), the behaviour is slightly contractant (\({\hbox {tr}}{\mathbf {D}}^0\approx 0.2\)). Note that \({\hbox {tr}}{\mathbf {D}}^0=0\) describes isochoric deformation and \({\hbox {tr}}{\mathbf {D}}^0=1\) applies for oedometric compression. In addition, the angle of dilatancy \(\psi\) is also shown in Fig. 5c.^{6} For an overconsolidation ratio of 2, clay is slightly dilatant (\({\hbox {tr}}{\mathbf {D}}^0\approx 0.1\)), for overconsolidated samples (OCR \(=6\)), \({\hbox {tr}}{\mathbf {D}}^0\approx 0.4\), cf. Fig. 5c. Arthur et al. [1] (cited in [33]) report that the angles of dilatancy \(\psi\) in the shear plane in dense biaxial tests with sand were about \(9^\circ \le \psi \le 30^\circ\). Simulations of overconsolidated samples (OCR \(=2\ldots 6\)) with barodesy result in angles of dilatancy in the range of \(3^\circ<\psi <14^\circ\), see Fig. 5c. As in this article, clay samples with arbitrary overconsolidation ratios are investigated, only a qualitative comparison of the values for \(\psi\) is possible.
5 Finite element calculations
State of the art in geotechnical engineering are calculations of stress and strain fields with finite element approaches. Commercial finite element programs often allow assessment of stability by means of socalled strength reduction analyses (\(\varphi\)–c reduction). A frequently used approach to assess stability is to reduce the shear parameters until loss of convergence in the numeric calculation. This approach is ambiguous, as convergence depends not only on the stability but also on numerical issues as, e.g. incrementation. Investigations on the occurrence of \({\hbox {tr}}{({\dot{{\mathbf {T}}}} {\mathbf {D}})}\le 0\) in finite element calculations could give a more clear identification of instability [17, 22, 25, 27]. The here presented finite element calculations have been performed using Abaqus. For Abaqus, a user subroutine Umat for the material model barodesy is available [31]. As barodesy is not formulated in the framework of elastoplasticity, a special strength reduction approach has been developed [32]. Secondorder work has been made available as additional output variable in the user material subroutine. Thus, secondorder work can be visualized easily in the abaqus framework for barodesy.
Note that in the finite element applications addressed here, \(W_2\) is evaluated with the actual \({\dot{{\mathbf {T}}}}\) and \({\mathbf {D}}\) tensors. In other words, a search for the \({\mathbf {D}}\)tensor that minimizes \(W_2\) has not been carried out, as it would render the calculations extremely lengthy. Consequently, the condition \(W_2=0\) could be encountered even earlier. An analytical solution for the \(W_2\) surfaces in stress space as developed by Niemunis [28] for hypoplasticity and applied by Meier et al. [22] is not yet archived for barodesy. However, the so obtained instability is still useful as indicator of failure.
5.1 Biaxial tests
The capability of modelling shear bands is an important property of material models. A first approach of visualizing shear bands in finite element calculations can be done on finemeshed biaxial tests [14]. Finite element calculations of biaxial tests with barodesy have already been performed by SchneiderMuntau et al. [31], and the appearance of shear bands has been discussed. The same example is used in this article for shear band visualization with the secondorder work criterion. For a biaxial test with a homogeneous void ratio distribution over 200 elements (\(e_{{\text {ini}}} = 0.55\)), all elements have the same deformation, see Fig. 6 for the stress–strain relationship and Fig. 7 for the secondorder work distribution. Secondorder work vanishes for all elements at the same calculation step, which corresponds to the peak at an axial strain of \(\varepsilon _1=6.7\%\). Note that in laboratory tests inhomogeneous deformation exists from the very beginning of the test, be it small or pronounced.
5.2 Slope stability
6 Conclusions
Vanishing secondorder work appears to be an suitable criterion for a situation where failure may occur. To evaluate \({\hbox {tr}}{(\dot{{\mathbf {T}}}}{\mathbf {D})}\), all possible \({\mathbf {D}}\)tensors and the pertinent \({\dot{{\mathbf {T}}}}\)tensors (resulting from a particular constitutive relation) should be investigated. This search is timeconsuming and has been applied for the element tests in Figs. 3, 4 and 5: we varied the stretching tensor in the deviatoric plane in order to search for minimum values of secondorder work. As soon as secondorder work vanishes, the investigated stress state belongs to the searched boundary. For a constant overconsolidation ratio, vanishing secondorder work is described by a cone. For normally consolidated to slightly overconsolidated soil, these cones lie inside the cone of critical stress states.
In barodesy, the secondorder work approach showed—like in hypoplasticity and elastoplasticity—that secondorder may vanish at stress states inside the critical limit surface. It is obtained that for rather loose soils, secondorder work vanishes inside the critical stress surface. Evaluating \(W_2=0\) in the nonconventional triaxial tests in Fig. 2 with the actual \({\mathbf {D}}\)tensor and with a variation of \({\mathbf {D}}\), almost led to the same results for mobilized friction angles.
 (1)
Straincontrolled drained biaxial test with an initial imperfection (a slightly higher void ratio in one element). The stress–strain behaviour is similar for all elements until the peak, but then gets chaotic. At a certain strain—corresponding to the strain at the peak of a homogeneous sample—an area of vanishing or negative secondorder work appears forming an endtoend shear band.
 (2)
Slope stability has been investigated by means of secondorder work. The first appearance of an endtoend band of \(W_2\le 0\) can define system failure.
Footnotes
 1.
Hypoplasticity and barodesy are constitutive models which do not use the standard notions of elastoplastic models (such as elastic region, yield function, flow function, ...). Instead, the effective corotated stress rate \({\mathring{{\mathbf {T}}}}\) is given as a function of stretching \({\mathbf {D}}\) and of effective stress \({\mathbf {T}}\), optionally complemented by further variables, as the void ratio e.
 2.
In this article, the simulations are carried out with barodesy for clay. All equations and a detailed description can be found in the open access article by Medicus and Fellin [20].
 3.
In this article, the mobilized friction angle when secondorder work vanishes is denoted as \(\varphi _{W_2}\).
 4.
It is common to define the overconsolidation ratio (OCR = \(p_{\text {e}}/p\)) by means of the socalled Hvorslev’s equivalent consolidation pressure \(p_{\text {e}}=\exp \left( \frac{N\ln (1+e)}{\lambda ^*}\right)\), divided by the actual mean stress p. \(p_{\text {e}}\) is the value of mean stress on the isotropic normal consolidation line which refers to the current specific volume \((1+e)\). If \(p/p_{\text {e}} =\) const in a deviatoric plane (\(p=\) const), then the void ratio e is constant.
 5.
From the earth pressure coefficient at rest \(K_0=1\sin \varphi _{\mathrm{c}}=\dfrac{1\sin \varphi _{\mathrm{m}}}{1+\sin \varphi _{\mathrm{m}}}\) we obtain with \(\varphi _{\mathrm{c}}=24^\circ\) the mobilized friction angle under oedometric conditions \(\varphi _{W_2}=\arcsin \dfrac{\sin \varphi _{\mathrm{c}}}{2\sin \varphi _{\mathrm{c}}}\approx 15^\circ\).
 6.
Note that the angle of dilatancy \(\psi\) in Fig. 5c strictly applies for axisymmetric compression (i.e. \(\alpha _\sigma =30^\circ\)) only. Experimental studies by Nakai [24] showed that \(0<\alpha _\sigma <15^\circ\) for plane strain conditions. These findings correspond to results obtained with barodesy [21].
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). Gertraud Medicus is grateful for the financial support by a research Grant of the Austrian Science Fund (FWF): P 28934N32. The authors thank Wolfgang Fellin for valuable discussion.
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