Modelling of shear localization in solids by means of energy relaxation
- Tuyet B Trinh^{1}Email author and
- Klaus Hackl^{1}
https://doi.org/10.1186/s40540-014-0009-0
© Trinh and Hackl; licensee Springer. 2014
Received: 27 October 2013
Accepted: 10 March 2014
Published: 12 May 2014
Abstract
An approach to the problem of shear localization is proposed. It is based on energy minimization principles associated with micro-structure developments. Shear bands are treated as laminates of first order. The micro-shear band is assumed to have a zero thickness, leading to an unbounded strain field and the special form of the energy within this micro-band. The energy is approximated by the mixture of potential of two low-strain and high-strain domains and it is non-convex. The problem of the non-convex energy arising due to the formation of shear bands is solved by energy relaxation in order to ensure that the corresponding problem is well-posed. An application of the proposed formulation to isotropic material is presented. The capability of the proposed concept is demonstrated through numerical simulation of a tension test.
Keywords
Energy relaxation Shear band Strain localizationBackground
Strain localization phenomena are observed in various materials as narrow zones of intense shearing, known as shear bands. In many cases, the formation of shear bands is accompanied by a softening response, characterized by a decrease in strength of the material with accumulated inelastic strain, often leading to complete failure [1],[2]. Therefore, research on formation of shear bands has received much attention.
In simulation of strain localization, mesh dependence is the direct consequence of the ill-posedness of the corresponding boundary value problem [3]. Some enhanced continuum approaches can be found in literatures such as Cosserat theory[4]-[6], nonlocal approaches[7],[8], and gradient-enhanced approach[9],[10]. In these, an internal length scale is introduced to reflect certain small-scale effects assumed to be present in shear bands. The disadvantage of the corresponding numerical models is, however, that the element size is required be at least an order of magnitude smaller than the width of shear zones in order to obtain results independent of the mesh size [11].
The strong discontinuity approach, known as an alternative way to simulate strain localization without the introduction of characteristic lengths, rests upon the assumption that the displacement field is discontinuous [12]-[14]. This approach can be categorized into unregularized and regularized strong discontinuities. For unregularized strong discontinuities, the discontinuous displacement field induces an unbounded strain field having the character of a Dirac-delta distribution [14]. For regularized strong discontinuity[11],[15], one considers a transition from continuous to discontinuous response by using an approximation of the Dirac-delta distribution. In both variants, however, it is necessary to determine the position of a shear band by tracking strong discontinuities.
Furthermore, another possibility to tackle the localization problem is the use of phenomenological plasticity frameworks, in which the shear band and its constitutive response are embedded in the macroscopic constitutive behavior. Pietruszczak and Xu [16],[17] suggested a theoretical framework for the analysis of brittle materials. Constituent materials including the intact and localized zone are used to determine the average mechanical properties through homogenization technique. The constitutive equation in the region confined by the shear band involves the resultant force rate acting at the interface and the displacement discontinuity. Amero [18] suggested a procedure for incorporating localized small-scale effects of the material response in the large-scale problem, which is characterized by the standard local continuum. The large-scale regularization of rate-dependent models is accomplished with the formalism of strong discontinuities to model effectively the localized dissipation observed in localized failures of solids and structures. Nguyen et al. [19],[20] presented an approach with enhanced kinematics to capture localized mode of deformation for quasi-brittle materials. The volume element intersected by a localization band is considered as a two-phase material. The continuity condition of the traction across the boundary of the localization boundary is enforced to couple two stresses corresponding to the behavior in the localization zone and the bulk elastic one.
In recent years a new methodology based on energy relaxation has been developed to simulate not only the development of material microstructures [21]-[28] but also localization phenomena in plasticity and damage [29]-[33]. For problems involving microstructure evolution and localization which is related to various local instability effects such as buckling, crashing, and cracking, integration of the stress-strain relation leads to a nonconvexity of the potential energy. This behaviour can be seen in many kinds of materials such as geomaterials, concrete, steel, composite. For detailed expositions of the different monotone stress-strain curves and the corresponding nonconvex energies consult [34]. Dacorogna [23] showed that minimizers cannot be obtained in nonconvex variational problems. Instead, the quasiconvex envelope of the nonconvex energy, called the relaxed energy, should be studied to ensure the existence of minimizers. For the problem of strain localization, shear bands are treated as laminates of first order in microscopic level. The advantage of this theory, when applied to the problem at hand, is the natural formation of shear bands based on the energy minimization principles associated with micro-structure developments. In the works of Miehe and his coworkers [29]-[31], the laminate orientation corresponding to a mode-II simple shear is approximated to the critical direction of non-convex energy based on the minimization of the determinant of the acoustic tensor. The width of a micro-shear band is finite. An incrementally variational formulation is based on an energy storage function and a dissipation function. Relaxation methods have been applied to crystal plasticity, see [25],[35], and the references therein. However, the model in this paper is different in the sense as the direction of the shear band is variable, while in crystal plasticity, it is fixed.
This approach has some similarities with that one of Miehe and coworkers [29]-[31]. There, a non-convex potential obtained as condensed energy of an incremental variational approach is used. This leads to microstructures given as laminates of finite width. In our approach, we start from an energy given as the minimum of a low strain and a high strain potential where the latter one has linear growth only, while Miehe’s energy has superlinear growth. This leads to degenerated laminates which can be interpreted as true shear bands.
This work is based on the formulation introduced in [36]. An application of the relaxation theory to linear elastic isotropic material and numerical simulations of a tension test under displacement control are shown. For inelastic materials, we assume that the elastic deformation is small compared to the inelastic deformation and can be neglected. A numerical example involving loading and unloading is studied in order to evaluate the performance of the proposed concept.
Existence of solutions of non-linear boundary value problems and relaxation
For elastic materials, this corresponds to the well-known principle of minimum of the potential energy. But inelastic materials can be incorporated as well via a time-incremental formulation. In this case, W denotes the so-called condensed energy [27],[28],[37].
Obviously the structure defined in (5), (9), (10) is completely analogous to that one given by (2), (1), (3) with u replaced by $\stackrel{\u0307}{u}$ and W(ε ) by $\Delta \left({\stackrel{\u0307}{\epsilon}}_{\mathrm{I}}\right)$.
From now on, we will focus our exposition on the elastic case keeping in mind that everything can be readily transferred to the inelastic case using the scheme explained above.
where ξ_{1} and ξ_{2}, respectively, are two volume fractions of the regions 1 and 2; ε_{1} and ε_{2}, respectively, are the strain fields within regions 1 and 2.
In Equation 17, or equivalently 18, we find the definition of the so-called first order lamination hull [23],[39]. This is nothing more than quasiconvexification restricted to first-order laminates as possible fluctuation fields. The formulation proposed in this paper is developed based on that very notion.
Shear bands as special laminates
α=2, where $\mathcal{D}$ is symmetric fourth-order, positive definite tensor. For α=1, this energy corresponds to a linear-elastic material with elastic stiffness tensor given by $\mathcal{D}$. For varying α, it behaves more or less stiff in a nonlinear way.
According to the assumption above, the potential inside a shear band is positive homogeneous of first-order in the strain field (19). We will see later on, that only for this very form of the potential as given in Equation 20 corresponding to α=2, it has the desired property leading to strong discontinuities. If α is smaller than 2, the material will exhibit only weak discontinuities. If α is larger than 2, a relaxed energy does not exist because of lack of coercivity.
Based upon these considerations, let us start with the consideration of a very simple one-dimensional model to discuss the physical implications of the proposed approach. Then it will be generalized to two dimensions.
One-dimensional problem
Micro-strain
Shear bands are treated as special laminates mixing two co-existed phases (Figure 3). The volume fraction ξ characterizing the width of the micro-band may be defined by the ratio between the length scale and a characteristic geometric parameter [30],[31]. If the volume fraction ξ is finite (0<ξ<1), shear bands are represented as weak discontinuity. In this case, let us denote by ε_{1} the strain present outside and by ε_{2} the strain present within a micro-shear band (see Figure 1a). A visualization given in Figure 1a depicts the shape of a non-convex potential W and its convexification.
The volume fraction ξ of the micro-band varies between 0 and 1. Let us assume that the volume fraction ξ is rather small in comparison to the volume fraction of the RVE, then, the latter case does not happen. If ξ tends to zero, the micro-strain ε_{2} of the high-strain domain is unbounded. Then, a potential W responsible for a strong discontinuity (ξ→0) is depicted in Figure1b.
The assumption of a zero-width micro-shear band immediately leads to an unbounded strain (27) within the high-strain domain of the micro-shear band.
Relaxed energy
where W^{mix} is refered to in the literature as the elasto-plastic superpotential [43]. In the plastic regime, the stress state lies on the yield surface, indicating that σ=A is in agreement with Equation 34.
Example 1
Now strains and stresses can be calculated as follows:
Two-dimensional problem
Micro-strain
where ${\mathit{\epsilon}}_{1}-{\mathit{\epsilon}}_{2}={(\mathit{a}\otimes \mathit{n})}^{\mathrm{s}}=\frac{1}{2}(\mathit{a}\otimes \mathit{n}+\mathit{n}\otimes \mathit{a})$ satisfies rank(ε_{1}−ε_{2})≤1 in Equation 18.
where ∥m ∥=1. Herein m and n are two unit vectors giving the direction of shear band evolution; s is an appropriately rescaled variable.
Relaxed potential
where $\mathcal{C}$ and $\mathcal{D}$ are symmetric fourth-order, positive definite tensors. Here, ε and γ are the strains in the low-strain and high-strain domains, respectively, whose domains are depicted in Figure 3. The energy W_{2}(γ ) is expected to be homogeneous of first order in γ , as shown in Figure 1b for the one-dimensional problem. Therefore, W_{2}(γ ) raised by the exponent of $\frac{1}{2}$ has the desired property leading to strong discontinuities. For convenience, $h=1\sqrt{N}/\mathit{\text{mm}}$ is introduced as a parameter to guarantee that W_{2}(γ ) has the dimension of energy density.
then, the mixture energy W^{mix} is the sum of ${W}_{1}^{\text{mix}}$ and ${W}_{2}^{\text{mix}}$.
Minimization problem: inf _{ s } W(s)
Expression | |
---|---|
Scalar minimization problem | infs W(s) |
Potential | W(s)=a s^{2}+b s+c|s| (c>0, a>0) |
Solution | $\underset{s}{inf}\phantom{\rule{0.3em}{0ex}}W\left(s\right)=-\frac{1}{4a}{(\left|b\right|-c)}_{+}^{2}$ |
Minimizer | $s=-\frac{1}{2a}{(\left|b\right|-c)}_{+}\phantom{\rule{0.3em}{0ex}}\text{sign}\left(b\right)$ |
Abbreviations | ${(\left|b\right|-c)}_{+}=\left\{\begin{array}{ll}0& \text{for}\left|b\right|\le c\\ \left|b\right|-c& \text{for}\left|b\right|>c\end{array}\right.$ |
$\text{sign}\left(b\right)=\frac{\left|b\right|}{b}\phantom{\rule{1em}{0ex}}\text{for}b\ne 0$ |
Herein $a=\frac{1}{2}\mathit{\gamma}:\mathcal{C}:\mathit{\gamma}$, $b=-\phantom{\rule{0.3em}{0ex}}\mathit{\epsilon}:\mathcal{C}:\mathit{\gamma}$, $c=h{\left(\mathit{\gamma}:\mathcal{D}:\mathit{\gamma}\right)}^{\frac{1}{2}}$. Easily one can recognise that a is positive due to the positive definiteness of the fourth-order tensor $\mathcal{C}$.
Computation of stress and the tangent operator
Localization criterion
As the process of deformation progresses, L may be negative, zero or positive. A positive value in turn signals the onset of localization, a criterion that can be shown to be equivalent to the well-known notion of loss of ellipticity:
L≤0: we have s=0. The relaxed potential W_{ R }(ε ) reduces to the elastic strain energy W_{1}(ε ).
L>0: we have s≠0. A shear band starts to develop. The homogeneous deformation ε decomposes into the two micro-strains ε_{1} and ε_{2}. The nonconvex potential energy W^{mix} is replaced with the approximated rank-one convexification W_{ R }(ε ) to ensure well-posedness of the problem.
Application of relaxation theory to linear isotropic material
Relaxed potential
In what follows, the case $\mathcal{D}$ being equal to $\mathcal{C}$ is investigated.
On inserting $\mathcal{D}=\mathcal{C}$ into (63) and (53), we obtain the mixture potential
where γ =(m ⊗n )^{ s }. Herein, $\mathcal{C}$ is the fourth-order isotropic elastic tensor
or in the tensor notation
Using the result of Equation 57 we obtain
The minimization of (74) with respect to s, ψ and φ yields
R≠0:
The capability of the proposed model is demonstrated through numerical simulation of a tension test in the next section.
Example 2
The displacement method is unable to capture the localization and shows a hardening behaviour as depicted in Figure 10a,b.
Application of relaxation theory to inelastic materials
Relaxed potential
For simplicity we consider the special case $\mathcal{D}={A}^{2}\mathcal{I}$[36] in this section with assuming orthogonality, i.e. m .n =0 of the two unit vectors giving the direction of shear band evolution. Herein, A is a material parameter and $\mathcal{I}$ is the fourth-order unity tensor. Furthermore, we assume that evolution of these two vectors over time is not remarkable.
then, the mixture dissipation potential Δ^{mix} is the sum of ${\Delta}_{1}^{\text{mix}}$ and ${\Delta}_{2}^{\text{mix}}$.
By substituting (1) into (83), the relaxed potential reads
The proposed formulation is implemented in the general code FEAP [45]. Based on the mixed enhanced strain method [46], the four-node quadrilateral element (MES element) will be considered in the next section.
Example 3
Conclusions
The paper focuses on a theoretical framework for the treatment of shear localization in solid materials. The theory is based on minimization principles associated with micro-structure developments under the assumptions of a micro-shear band of a zero thickness and the presence of a mixture potential inside the shear band.
Localization phenomena are regarded as micro-structure developments associated with nonconvex potentials. The nonconvexity of the mixture potential occurring due to the formation of strain localization is resolved by relaxation in order to ensure the well-posedness of the associated boundary value problem. The relaxed potential, which is approximated by a first-order rank-one convexification, is obtained via local minimization problem of the mixture potential. The onset of localization is detected through the proposed optimization process. The model can be applied to any material which softens towards the critical state. The relaxed stress can be computed directly and approaches the critical stress as soon as strain localization occurs. Material points located inside the shear bands can be considered as decomposed into a low strain and a high strain phase at the microscopic level. The theoretical solutions satisfy possessing a zero micro-band width at the microscopic level. At the macroscopic scale, the width of a shear band is still finite. Numerical results clearly show a mesh-independent behaviour in the sense that shear bands are as narrow as the mesh resolution allows, while all other features of the solution are independent of the chosen discretization.
Declarations
Acknowledgements
The research was supported through grants by the Vietnamese Government and Ministry of Education and Training as well as the Research School at Ruhr University Bochum, Germany.
Authors’ Affiliations
References
- Borja R: A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation. Comput Methods Appl Mech Engrg 2000, 190: 1529–1549. 10.1016/S0045-7825(00)00176-6View ArticleGoogle Scholar
- Lott-Crumpler DA: The formation and properties of shear bands in viscoplastic materials. Model Simul Mater Sc 1997, 5: 317–336. 10.1088/0965-0393/5/4/002View ArticleGoogle Scholar
- Engelen RAB, Geers MGD, Baaijens FPT: Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int J Plasticity 2003, 19: 403–433. 10.1016/S0749-6419(01)00042-0View ArticleGoogle Scholar
- Mühlhaus HB, Vardoulakis I: The thickness of shear bands in granular materials. Géotechnique 1987, 37: 271–283. 10.1680/geot.1987.37.3.271View ArticleGoogle Scholar
- de Borst R, Sluys LJ: Localization in a Cosserat continuum under static and dynamics loading conditions. Comput Methods Appl Mech Engrg 1991, 90: 805–827. 10.1016/0045-7825(91)90185-9View ArticleGoogle Scholar
- Huang W, Bauer E: Numerical investigations of shear localization in a micro-polar hypoplastic material. Int J Numer Anal Methods Geomech 2003, 27: 325–352. 10.1002/nag.275View ArticleGoogle Scholar
- Strömberg L, Ristinmaa M: FE-formulation of a non-local plasticity theory. Comput Methods Appl Mech Engrg 1996, 136: 127–144. 10.1016/0045-7825(96)00997-8MathSciNetView ArticleGoogle Scholar
- Baz̆ant ZP, Belytschko TB, Chang TP: Continuum model for strain softening. J Eng Mech (ASCE) 1984, 110: 1666–1692. 10.1061/(ASCE)0733-9399(1984)110:12(1666)View ArticleGoogle Scholar
- de Borst R, Mühlhaus HB: Gradient-dependent plasticity: formulation and algorithmic aspects. Int J Numer Meth Eng 1992, 35: 521–539. 10.1002/nme.1620350307View ArticleGoogle Scholar
- Pamin J: Gradient-dependent plasticity in numerical simulation of localization phenomena. Delft University of Technology, The Netherlands; 1994.Google Scholar
- Larsson R, Steinmann P, Runesson K: Finite element embedded localization band for finite strain plasticity based on a regularized strong discontinuity. Mech Cohes-Frict Mat 1998, 4: 171–194. 10.1002/(SICI)1099-1484(199903)4:2<171::AID-CFM81>3.0.CO;2-XView ArticleGoogle Scholar
- Simo JC, Oliver J, Armero F: An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 1993, 12: 277–296. 10.1007/BF00372173MathSciNetView ArticleGoogle Scholar
- Oliver J: Modelling strong discontinuities in solid mechanics via strain softening constitutive equations: fundamental and numerical simulation. Monograph CIMNE No. 28. Internacional Center for Numerical Methods in Engineering, Barcelona; 1995.Google Scholar
- Armero F, Garikipati K: An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int J Solids Struct 1996, 33: 2863–2885. 10.1016/0020-7683(95)00257-XMathSciNetView ArticleGoogle Scholar
- Steinmann P, Larsson R, Runesson K: On the localization properties of multiplicative hyperelasto-plastic continua with strong discontinuities. Int J Solids Struct 1997, 34: 969–990. 10.1016/S0020-7683(96)00043-1View ArticleGoogle Scholar
- Pietruszczak S, Xu G: Brittle response of concrete as a localization problem. Int J Solids Struct 1995, 32: 1517–1533. 10.1016/0020-7683(94)00231-KView ArticleGoogle Scholar
- Xu G, Pietruszczak S: Numerical analysis of concrete fracture based on a homogenization technique. Comput Struc 1997, 63: 497–509. 10.1016/S0045-7949(96)00304-5View ArticleGoogle Scholar
- Armero F: Large-scale modeling of localized dissipative mechanisms in a local continuum: applications to the numerical simulation of strain localization in rate-dependent inelastic solids. Mech Cohes-Frict Mater 1999, 4: 101–131. 10.1002/(SICI)1099-1484(199903)4:2<101::AID-CFM78>3.0.CO;2-YView ArticleGoogle Scholar
- Nguyen GD, Einav I, Korsunsky AM: How to connect two scales of behaviour in constitutive modelling of geomaterials. Geotech Lett 2012, 2: 129–134. 10.1680/geolett.12.00030View ArticleGoogle Scholar
- Nguyen GD, Einav I, Korsunsky AM: A constitutive modelling framework featuring two scales of behaviour: Fundamentals and applications to quasi-brittle failure. Eng Fract Mech 2012, 115: 221–240. 10.1016/j.engfracmech.2013.11.006View ArticleGoogle Scholar
- Hackl K, Hoppe U (2001) On the calculation of microstructures for inelastic materials using relaxed enegies. In: Miehe C (ed)Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, 77–86, Boston. Hackl K, Hoppe U (2001) On the calculation of microstructures for inelastic materials using relaxed enegies. In: Miehe C (ed)Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, 77–86, Boston.Google Scholar
- Ball JM, James RD: Fine phase mixtures as minimizer of energy. Arch Ration Mech Ana 1987, 100: 13–52. 10.1007/BF00281246MathSciNetView ArticleGoogle Scholar
- Dacorogna B: Direct methods in the calculus of variations. Springer-Verlag, Berlin; 1989.View ArticleGoogle Scholar
- Kohn R: The relaxation of a double-well energy. Continuum Mech Therm 1991, 3: 193–236. 10.1007/BF01135336MathSciNetView ArticleGoogle Scholar
- Bartel S, Carstensen C, Hackl K, Hoppe U: Effective relaxation for microstructure simulations: algorithms and applications. Comput Methods Appl Mech Engrg 2004, 193: 5143–5175. 10.1016/j.cma.2003.12.065MathSciNetView ArticleGoogle Scholar
- Hackl K, Heinen R: A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy. Continuum Mech Therm 2008, 19: 499–510. 10.1007/s00161-008-0067-zMathSciNetView ArticleGoogle Scholar
- Carstensen C, Hackl K, Mielke A: Non-convex potentials and microstructures in finite-strain plasticity. Proc R Soc Lond A 2002, 458: 299–317. 10.1098/rspa.2001.0864MathSciNetView ArticleGoogle Scholar
- Hackl K, Fischer FD: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc R Soc Lond A 2008, 464: 117–132. 10.1098/rspa.2007.0086MathSciNetView ArticleGoogle Scholar
- Lambrecht M, Miehe C, Dettmar J: Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic-plastic bar. Int J Solids Struct 2003, 40: 1369–1391. 10.1016/S0020-7683(02)00658-3View ArticleGoogle Scholar
- Miehe C, Lambrecht M: A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solids: small-strain theory for standard dissipative materials. Comput Methods Appl Mech Engrg 2003, 192: 473–508. 10.1016/S0045-7825(02)00547-9MathSciNetView ArticleGoogle Scholar
- Miehe, C, Lambrecht, M: Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative solids Int J Numer Meth Eng 2003, 58: 1–41. 10.1002/nme.726MathSciNetView ArticleGoogle Scholar
- Miehe C, Lambrecht M, Guerses E: Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative solids. J Mech Phys Solids 2004, 52: 2725–2769. 10.1016/j.jmps.2004.05.011MathSciNetView ArticleGoogle Scholar
- Guerses E, Miehe C: On evolving deformation microstructures in non-convex partially damaged solids. J Mech Phys Solids 2011, 59: 1268–1290. 10.1016/j.jmps.2011.01.002MathSciNetView ArticleGoogle Scholar
- Mistakidis ES: On the solution of structures involving elements with nonconvex energy potentials. Struct Optimization 1997, 13: 182–190. 10.1007/BF01199238View ArticleGoogle Scholar
- Kochmann DM, Hackl K: The evolution of laminates in finite plasticity: a variational approach. Continuum Mech Thermodyn 2011, 23: 63–85. 10.1007/s00161-010-0174-5MathSciNetView ArticleGoogle Scholar
- Trinh BT, Hackl K: Finite element simulation of strain localization in inelastic solids by energy relaxation. Vietnam J Mech 2011, 33: 203–213.Google Scholar
- Mielke A: Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Comput Methods Appl Mech Engrg 2004, 193: 5095–5127. 10.1016/j.cma.2004.07.003MathSciNetView ArticleGoogle Scholar
- de Borst R: Damage, material instabilities, and failure, Chapter 10. In Encyclopedia of Computational Mechanics, vol. 2. Edited by: Stein E, de Borst R, Hughes TJR. Wiley, Chichester; 2004:335–373.Google Scholar
- Kohn RV, Strang G: Optimal design and relaxation of variational problems I, II, III. Comm Pure Appl Math 1986, 39: 113–137. 10.1002/cpa.3160390107MathSciNetView ArticleGoogle Scholar
- Hencky H: Zur Theorie plastischer Deformation und der hierdurch im Material hervorgerufenen Nachspannungen. Z Angew Nath Mech 1924, 4: 323–333. 10.1002/zamm.19240040405View ArticleGoogle Scholar
- Ilyushin AA: Some problems of plastic deformations. PMM 1943, 7: 245–272.MathSciNetGoogle Scholar
- Ilyushin AA: Theory of plasticity at simple loading of the bodies exhbiting plastic hardening. PMM 1947, 11: 291–296.MathSciNetGoogle Scholar
- Gao Y, Huang K: Complementary energy principles for elasto-perfectly plasticity. Scientia Sinica 1988, 31: 1469–1476.MathSciNetGoogle Scholar
- Budhu M: Soil mechanics and foundations. United States of America, Wiley; 2000.Google Scholar
- Taylor RL (2004) A finite element analysis program: User manual, University of California at Berkeley. Taylor RL (2004) A finite element analysis program: User manual, University of California at Berkeley.Google Scholar
- Kasper EP, Taylor RL (1997) A mixed-enhanced strain method: linear problems. Report No. UCB/SEMM-97/02, University of California at Berkeley. Kasper EP, Taylor RL (1997) A mixed-enhanced strain method: linear problems. Report No. UCB/SEMM-97/02, University of California at Berkeley.Google Scholar
- Zienkiewicz OC, Taylor RL: The finite element method. Volume 1: The Basis (5nd edn). Butterworth-Heinmann, London; 2000.Google Scholar
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