Modelling of shear localization in solids by means of energy relaxation
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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.
Notes
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.
Supplementary material
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