Numerical technique for strain localization analysis considering a Cartesian parameterization

Abstract

This paper presents a numerical technique for strain localization analysis in nonlinear material models considering a Cartesian parameterization. As a result of material’s natural heterogeneity, degradation often occurs in a small and weaker portion of the body. This concentration of irreversible phenomena is commonly referred as strain localization. From a kinematic standpoint, strain localization is associated with weak discontinuities that occur during physically nonlinear structural analysis. In a numerical simulation, it is linked with the loss of ellipticity of differential equations governing the boundary value problem. Singularity of the acoustic tensor is considered the classical condition for strain localization. It can be approached via analytical formulations or numerical techniques. Such a parameterization was utilized to define the normal direction to the discontinuity surface. Localization analysis was performed at material level after the convergence of each step in a set of nonlinear analyses. After the simulations, valuable information is available to regularization methods.

Appendices

Appendix 1

The face of the parameterized space where z is constant indicates that ${\mathbf{v}} = [x, y, 1]^\mathrm{T}$. At this face, each component of the acoustic tensor is given by:

$$\begin{aligned} Q_{11}&= x^2 D_{1111} + xy D_{1112} + x D_{1113} + xy D_{2111} + y^2 D_{2112} \\&\quad + y D_{2113} + x D_{3111} + y D_{3112} + D_{3113} \\ Q_{12}&= x^2 D_{1121} + xy D_{1122} + x D_{1123} + xy D_{2121} + y^2 D_{2122} \\&\quad + y D_{2123} + x D_{3121} + y D_{3122} + D_{3123} \\ Q_{13}&= x^2 D_{1131} + xy D_{1132} + x D_{1133} + xy D_{2131} + y^2 D_{2132} \\&\quad + y D_{2133} + x D_{3131} + y D_{3132} + D_{3133} \\ Q_{21}&= x^2 D_{1211} + xy D_{1212} + x D_{1213} + xy D_{2211} + y^2 D_{2212} \\&\quad + y D_{2213} + x D_{3211} + y D_{3212} + D_{3213} \\ Q_{22}&= x^2 D_{1221} + xy D_{1222} + x D_{1223} + xy D_{2221} + y^2 D_{2222} \\&\quad + y D_{2223} + x D_{3221} + y D_{3222} + D_{3223} \\ Q_{23}&= x^2 D_{1231} + xy D_{1232} + x D_{1233} + xy D_{2231} + y^2 D_{2232} \\&\quad + y D_{2233} + x D_{3231} + y D_{3232} + D_{3233} \\ Q_{31}&= x^2 D_{1311} + xy D_{1312} + x D_{1313} + xy D_{2311} + y^2 D_{2312} \\&\quad + y D_{2313} + x D_{3311} + y D_{3312} + D_{3313} \\ Q_{32}& = x^2 D_{1321} + xy D_{1322} + x D_{1323} + xy D_{2321} + y^2 D_{2322} \\&\quad + y D_{2323} + x D_{3321} + y D_{3322} + D_{3323} \\ Q_{33}& = x^2 D_{1331} + xy D_{1332} + x D_{1333} + xy D_{2331} + y^2 D_{2332} \\&\quad + y D_{2333} + x D_{3331} + y D_{3332} + D_{3333} \end{aligned}$$

(7)

Recalling Eq. 6, the objective function f is given by:

$$\begin{aligned} f&= {\text {det}}(\mathbf{Q} ) = Q_{11} Q_{22} Q_{33} + Q_{12} Q_{23} Q_{31} + Q_{13} Q_{21} Q_{32} \\&\quad - (Q_{13} Q_{22} Q_{31} + Q_{11} Q_{23} Q_{32} + Q_{12} Q_{21} Q_{33}) \end{aligned}$$

(8)

Applying the components explicitly derived in Eq. 7 into Eq. 8 and grouping them accordingly with their polynomial degree, one obtains:

$$\begin{aligned} f(x,y)&= {\text {det}}{} \mathbf{Q} (x,y) = a + bx + cy + dx^2 + exy + fy^2 \\&\quad + gx^3 + h x^2y + l xy^2 + m y^3 + n x^4 + o x^3y \\&\quad + p x^2y^2 + q xy^3 + r y^4+ s x^5 + t x^4y + u x^3y^2 \\&\quad + v x^2y^3 + w xy^4 + ab y^5 + ac x^6 + ad x^5y \\&\quad + ae x^4y^2 + af x^3y^3 + ag x^2y^4 + ah xy^5 + al y^6 \end{aligned}$$

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where $a, b, c, d, \ldots , ag, ah, al$ are coefficients of a polynomial expression computed directly from terms of the tangent constitutive tensor.

Newton’s iterative method is employed to compute the pair $(x_{\mathrm{sol}}, y_{\mathrm{sol}})$ that minimizes f. A random point within the parameterized space is selected to be the start point of the method.

Given the definition of Newton’s method, an algebraic expression for the objective function f and its first- and second-order derivatives is required. Following the polynomial notation defined in Eq. 9, the derivatives are obtained in a direct manner:

$$\begin{aligned} \frac{\partial f}{\partial x}& = b + 2dx + ey + 3gx^2 + 2hxy + ly^2 + 4nx^3 \\&\quad + 3ox^2y + 2pxy^2 + qy^3 + 5sx^4 + 4tx^3y \\&\quad + 3ux^2y^2+ 2vxy^3 + wy^4 + 6acx^5 + 5adx^4y \\&\quad+ 4aex^3y^2 + 3afx^2y^3 + 2agxy^4 + ahy^5 \end{aligned}$$

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$$\begin{aligned} \frac{\partial f}{\partial y}&= c + ex + 2fy + hx^2 + 2lxy + 3my^2 + ox^3 \\&\quad + 2px^2y + 3qxy^2 + 4ry^3 + tx^4 + 2ux^3y \\&\quad + 3vx^2y^2 + 4wxy^3 + 5aby^4 + adx^5 + 2aex^4y \\&\quad + 3afx^3y^2 + 4agx^2y^3 + 5ahxy^4 + 6aly^5 \end{aligned}$$

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Appendix 2

1.1 Isotropic damage models

An exponential damage evolution law was adopted for Mazars [24] and Simo and Ju [44]:

$$\begin{aligned} D(\varepsilon _{\mathrm{eq}}) = 1 - \frac{\kappa _0}{\varepsilon _{\mathrm{eq}}} [1 - \alpha + \alpha e^{-\beta (\varepsilon _{\mathrm{eq}} - \kappa _0)}] \end{aligned}$$

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where $\varepsilon _{\mathrm{eq}}$ is the equivalent deformation calculated accordingly for each constitutive model; $\kappa _0$ is the equivalent deformation after which the damage is initiated; $\alpha$ is the maximum admissible damage; $\beta$ is the intensity of damage evolution.

Parameters $\alpha _\mathrm{t}$ and $\alpha _\mathrm{c}$, presented in Table 1 for Mazars [24] isotropic damage model, are weight functions to account for different material behaviors under tensile and compressive stresses. Within the model’s formulation, they are used to compute material degradation (D) as follows:

$$\begin{aligned} D = \alpha _\mathrm{t} D_\mathrm{t} + \alpha _\mathrm{c} D_\mathrm{c} \end{aligned}$$

(13)

where $D_\mathrm{t}$ and $D_\mathrm{c}$ represent damage from tensile and compressive loads, respectively.

For specific information about this constitutive model, the reader is referred to [24].

1.2 Smeared cracking constitutive model

Parameters presented in Table 1 correspond to:

• $f_{\mathrm{t}}$: Tensile strength

• $f_{\mathrm{c}}$: Compressive strength

• $\varepsilon _{\mathrm{c}}$: Compressive strain limit

• $G_{\mathrm{f}}$: Energy release rate for fracture

• h: Characteristic length

• $\beta _\mathrm{r}$: Shear retention factor

1.3 von Mises plasticity

Parameters presented in Table 1 correspond to:

• $\sigma _\mathrm{y}$: Yield strength

• H: Hardening modulus

Appendix 3

INSANE (INteractive Structural ANalysis Environment) platform is an open-source project based on the Java language and developed in the Structural Engineering Department of the Federal University of Minas Gerais (UFMG). Figure 2 presents an overview of INSANE’s numerical core. Parts of an XML input file, interpreted by Persistence class, are presented and described in this appendix.

Listing 1 XML input file for Solution class

Listing 1 provides parameters for the objects of Solution class, including the number of steps (200), maximum number of iterations for each step (15) and numerical method for solving each step of the nonlinear structural analysis (standard Newton–Raphson).

It also specifies the control method used during the incremental-iterative solution process. For this example, the displacement control was utilized considering node #7 in the horizontal direction (x).

Listing 2 XML input file for Model class

Listing 2 provides parameters for the objects of Model class, including properties of nodes (e.g., coordinates, restraints and prescribed displacements), materials (e.g., elasticity modulus and Poisson’s ratio) and elements (e.g., incidence). The model type (FEM, finite element method), problem driver (physically nonlinear analysis) and analysis model (plane stress) are also specified in this XML excerpt. Special focus is given to Boolean parameter LocalizationAnalysis: It triggers the localization analysis described in this paper after convergence of each step of the nonlinear structural analysis.