Large deformation analysis of plane-stress hyperelastic problems via triangular membrane finite elements
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Abstract
A finite-element formulation based on triangular membranes of any order is proposed to analyze problems involving highly deformable hyperelastic materials under plane-stress conditions. The element kinematics is based on positional description and the degrees of freedom are the current plane coordinates of the nodes. Two isotropic and nonlinear hyperelastic models have been selected: the compressible neo-Hookean model and the incompressible Rivlin–Saunders model. The constitutive relations and the consistent tangent operator are condensed to the compact 2D forms imposing plane-stress conditions. The resultant algorithm is implemented in a computer code. Three benchmark problems are numerically solved to assess the formulation proposed: the Cook’s membrane, involving bending, shear, and a singularity point; a partially loaded membrane, which presents severe mesh distortion and large compression levels; and a rubber sealing, which is a more realistic problem. Convergence analysis in terms of displacements, applied forces, and stresses is performed for each problem. It is demonstrated that mesh refinement avoids locking problems associated with incompressibility condition, bending-dominated problems, stress concentration, and mesh distortion. The processing times are relatively small even for fifth-order elements.
Keywords
Isoparametric triangular finite elements of any order Hyperelasticity Large deformation analysis Plane-stress conditionIntroduction
The Finite-Element Method (FEM) has been widely employed to solve general problems in all branches of engineering. An important application of the FEM is the prediction of the mechanical behavior of structures. There are many types of finite elements in the context of structural engineering: bars (trusses, beams, and frames), two-dimensional elements (membrane, plates, and shells), and three-dimensional (solid) elements. A very simple example is the 2D membrane finite element with triangular geometry, which is the topic of the present paper. This element has been extensively used in plane structural analyses, that is, in predicting the mechanical behavior of structures under plane-stress and plane strain conditions. Such structures are very common in civil, mechanical, and aeronautic engineering, for instance.
Finding the most appropriate finite-element formulation for specific applications remains a major challenge. It is always desirable that the formulation be robust, accurate, reliable and, at the same time, simple. The standard FEM in structural mechanics is displacement-based, i.e., the unknown quantities (or degrees of freedom) are nodal displacements and, in some cases, nodal rotations. The equilibrium of forces is then described via a single-field variational principle, equivalent to Principle of Virtual Work. The strains and stresses are computed from the exact compatibility equations and constitutive laws, respectively. Despite the simplicity of the standard formulation, several alternative techniques have been developed to improve the overall performance of the elements. For example, it is often said that standard 2D finite elements have a poor performance in structural problems involving bending, domains with complex geometries, and near-incompressibility regime. According to Angoshtari et al. (2017), these drawbacks can be overcome using mixed methods, in which the stress field is usually an independent variable and, thus, can be computed with higher accuracy when compared to the standard formulation. An example of mixed formulation is the Enhanced Assumed Strain (EAS) method developed by Simo and Rifai (1990), in which the strain field is enriched by adding some incompatible modes. However, mixed methods are considerably more complicated than the standard formulation, increasing the number of degrees of freedom. Another shortcoming of mixed EAS methods is the presence of unphysical (hourglass) instabilities in compression problems. In most cases, the use of an hourglass stabilization technique is required, increasing the complexity of the analysis. Moreover, as pointed out by Jabareen and Rubin (2014), mixed triangular elements may provide an unrealistic soft response in bending-dominated problems.
The element order of approximation considerably affects the numerical results obtained in structural analyses using the FEM. It is well known that high-order elements provide more accurate results, but, on the other hand, demand more computational effort in terms of memory capacity and processing time. Linear order elements, in turn, are very easy to be implemented, simple to be integrated over the domain and provide a fast solution, but, in general, exhibit a very stiff behavior (locking phenomenon) even with an extremely refined mesh. A better performance is usually obtained with triangular elements of quadratic order, but some authors argue that the use of these elements together with full integration provides inaccurate results for nearly incompressible materials (Jabareen and Rubin 2014). Besides, to decrease the processing time for higher order elements, reduced and selective integration techniques have been proposed. However, these methods are complex and case-dependent. For the present author, the accuracy of results when using fully integrated high-order elements is always assured and the large computational effort required can be circumvented by adopting high-performance computers and parallel processing techniques. Following this line of reasoning, the present finite element is the isoparametric triangular membrane element of any order based on positional description and fully integrated. In other words, the order of approximation is general (linear, quadratic, cubic, etc.) and the degrees of freedom are current nodal positions (instead of displacements). High-order elements fully integrated and based on positional FEM have been successfully applied to solve general structural problems in the works of Pascon and Coda (2012, 2013, 2015, 2017), for instance. It has been demonstrated, in these works, that the accuracy of results is guaranteed for sufficiently refined meshes.
The development of finite-element formulations capable of modeling highly deformable materials has received considerable attention over the last decades. In the case of finite displacements, the geometrically nonlinear analysis is imperative, as the change of configuration may be large. In other words, the equilibrium of forces must be achieved at the final deformed configuration, which leads to a nonlinear system of equations regarding the degrees of freedom. However, this system can be solved using standard numerical methods, as the Newton–Raphson iterative technique, which is widely adopted in nonlinear analyses (see, for example, the work of Crisfield 2000).
An example of highly deformable engineering material is the elastomer, which may present large levels of elastic (or reversible) strains. Numerous applications of elastomeric materials can be cited, including bridge bearings, engine parts, vehicle door seals, o-rings, tires, wire insulation, industrial belts, polyurethane foams for car seats, etc. In the case of finite elastic strains, the material behavior is described by hyperelastic constitutive models. The response of a hyperelastic material is expressed via a scalar potential energy density called Helmholtz free-energy function, which must satisfy the following conditions: normality, polyconvexity, and coercivity. The first condition means that the energy is zero when there is no strain. The second requirement is necessary to obtain a unique solution for a given strain or stress field. The third condition is used to prevent the material to be annihilated (the energy tends to infinity when the volume tends to zero) or stretched into an infinite range (the energy must also become infinity when the volume tends to infinity or the strain levels becomes infinitely large). One should note that the linear elastic models, as the Hooke and Saint–Venant–Kirchhoff laws, do not satisfy the polyconvexity and coercivity requirements and, thus, cannot be used to reproduce large levels of elastic strain. Further details regarding hyperelasticity and comparison of the hyperelastic models proposed in the scientific literature can be found, for instance, in the works of Holzapfel (2000) and Pascon (2008).
For general hyperelastic materials, the stress tensor is obtained by differentiating the Helmholtz free-energy function with respect to the strain tensor, which leads to a nonlinear 3D constitutive law; that is, the expressions to obtain the stresses in terms of the strains are nonlinear. In the specific case of plane-stress conditions, a relationship among the strains appears and, thus, the 3D model is condensed into a compact 2D form according to the null stresses. For linear elastic models, the expressions to determine the plane-stresses and the normal out-of-plane strain are all explicit. However, the condensation of the 3D model for plane-stress conditions is not straightforward for nonlinear hyperelastic laws, as the resultant expressions employed to compute stresses in terms of strains are implicit. Two isotropic hyperelastic models are adopted in the present paper: the compressible neo-Hookean law (nH), employed in Sze et al. (2004), for instance; and the incompressible Rivlin–Saunders model (RS), initially proposed by Rivlin (1956) and slightly modified in Pascon (2008). Both models have been extensively employed to reproduce the mechanical response of elastomers, presenting good agreement with experimental data. In the first case (nH), an implicit and nonlinear relationship among strains must be solved to obtain the out-of-plane strain component. With this component, the plane stresses can be obtained via explicit expressions. For the second model (RS), the strategy of the Lagrangian multiplier is adopted, in which explicit nonlinear expressions are employed to obtain the plane stresses. Although this strategy is well known in the scientific literature, little discussion concerning the consistent tangent operator has been made. Some details regarding this operator, widely employed in nonlinear analyses, are provided in the present paper.
The purpose of this work is to develop a finite-element formulation with isoparametric triangular elements for large deformation analysis of nonlinear hyperelastic materials under plane-stress state. The paper is organized as follows. In “Finite element approximation” section, the finite-element kinematics for triangular elements of any order is described. In “Constitutive modeling” section, the constitutive models employed, as well as the expressions involved in the case of plane-stress conditions, are provided. The equilibrium principle and the numerical algorithm adopted, which are valid for other types of plane elements, are given in “Numerical algorithm” section. The validating numerical examples involving highly deformable materials under plane-stress state are described in “Numerical examples” section. Finally, the concluding remarks drawn from the examples are provided in “Conclusions” section.
Finite-element approximation
Expressions (1) and (2) can be employed for any order of approximation (linear, quadratic, cubic, etc.). To determine the shape functions for any degree of approximation, the algorithm developed in Pascon and Coda (2013) is adopted here. This general strategy is used to calculate—based on the element order—the number and the non-dimensional coordinates of the nodes, the polynomial coefficients of the shape functions, and their values at any point inside the element domain. Such algorithm is extremely convenient if a finite-element formulation of any order is to be implemented in a computer code, as there is no need to set and write all the shape function coefficients for various orders of approximation, which is very cumbersome especially for high orders.
Constitutive modeling
Compressible neo-Hookean model (nH)
It should be emphasized that the stretch component C_{33} depends on the plane components; that is, C_{33} is a function of the set \( \left( {C_{11} ,C_{12} ,C_{22} } \right) \).
The method to determine the component C_{33} from the plane stretches is provided in “Numerical algorithm” section.
Similar expressions for the plane-stress nH model can be found in the work of Kirchner et al. (1997) considering the left Cauchy–Green stretch tensor (\( {\mathbf{B}} = {\mathbf{FF}}^{T} \)), the Kirchhoff stress tensor (\( {\varvec{\uptau}} = {\mathbf{FSF}}^{T} \)), and logarithmic stretches. However, the resultant model does not describe the same material response of the present constitutive law because of the different measures adopted in each work.
Incompressible Rivlin–Saunders model (RS)
One should remember that both expressions (15) and (27) are plane compact forms, i.e., the matrices that appear have dimensions 2 × 2.
Numerical algorithm
The component C_{33} is iteratively updated until r_{C33} is sufficiently small.
The derivatives in terms of tensor C are given in “Appendix 2”. The dimensions of the consistent tangent operators (39) and (43) are 2 × 2 × 2 × 2.
Numerical examples
In this section, the performance of the finite-element formulation developed is assessed. To this end, three structural problems involving highly deformable hyperelastic materials under plane-stress conditions are numerically analyzed: the Cook’s membrane, a partially loaded membrane, and a rubber sealing.
The formulation proposed is implemented in a computational code written in FORTRAN language and, thus, the numerical results are obtained from computer simulations. To this end, a parallel processing technique in a cluster with 12 processors has been used. Comparison with numerical results from the scientific literature is performed to validate the present methodology. The tolerance adopted for the errors (36) and (40) in all the examples is 10^{−6}.
Several meshes are employed for all the examples to analyze the influence of the mesh refinement on the accuracy of results. Both hierarchical and polynomial refinements are used, that is, the number and the order of the elements are increased until there is convergence of results. To perform this convergence analysis, all the meshes are generated from the same base mesh. The nodes are equally spaced for each quadrilateral of the base mesh.
In most of the finite-element papers, only the displacement fields are analyzed. However, the stress level control is also extremely important for design purposes. The stress values at the nodes are calculated in a post-processing code. After simulating the structure with the main program, the approximated displacement fields at the end of some specific steps are stored. Based on these fields, which are represented by the vector of degrees of freedom, one can determine strains and stresses at any point inside the domain. Since a general node can belong to various elements, the stress values at this node obtained for each element may be different. Therefore, the stresses are determined previously at the numerical integration points and then transferred to the nodes via a linear interpolation based on the least-squares method. For the nodes that belong to more than one element, a simple average of the nodal stresses is used. It should be said that another numerical strategies for stress field determination could be employed and compared to the present one.
Cook’s membrane
Three material models have been selected to analyze the present example, according to Fig. 2. The vertical load reaches the value of 40 N/mm for the nH model and 0.4 N/mm for the other two models. The coefficients of the nH model are the same as those adopted in Angoshtari et al. (2017) and Sze et al. (2004) for a generic near-incompressible material. The use of a bulk modulus K much larger than the shear modulus μ leads to a material model in which the volumetric strength is high and, thus, the volume changes are small. The material coefficients for the MR and BBC models have been interpolated in the work of Pascon (2008) from the experimental data of Yeoh (1997) for unfilled natural rubber vulcanizates, considering the incompressibility condition.
Convergence analysis regarding displacements for the nH model
ORD | NDOF | NE | NIP | u _{1} | u _{2} | PT (s) | TNI |
---|---|---|---|---|---|---|---|
1 | 24 | 12 | 1 | − 11.33 | 19.69 | 1 | 300 |
70 | 48 | − 20.26 | 24.17 | 1 | 300 | ||
234 | 192 | − 25.24 | 25.55 | 1 | 301 | ||
850 | 768 | − 27.18 | 25.97 | 1 | 300 | ||
1850 | 1728 | − 27.62 | 26.06 | 2 | 300 | ||
3234 | 3072 | − 27.79 | 26.11 | 4 | 300 | ||
5002 | 4800 | − 27.88 | 26.13 | 8 | 300 | ||
7154 | 6912 | − 27.93 | 26.15 | 11 | 311 | ||
2 | 70 | 12 | 12 | − 24.87 | 25.56 | 1 | 313 |
234 | 48 | − 27.55 | 26.00 | 1 | 311 | ||
494 | 108 | − 27.88 | 26.10 | 1 | 306 | ||
850 | 192 | − 27.96 | 26.13 | 2 | 302 | ||
1302 | 300 | − 28.00 | 26.15 | 3 | 300 | ||
1850 | 432 | − 28.03 | 26.17 | 5 | 300 | ||
3 | 140 | 12 | 13 | − 27.60 | 25.91 | 1 | 339 |
494 | 48 | − 27.99 | 26.14 | 1 | 336 | ||
1064 | 108 | − 28.05 | 26.19 | 2 | 347 | ||
1850 | 192 | − 28.09 | 26.21 | 4 | 353 | ||
2852 | 300 | − 28.11 | 26.22 | 8 | 359 | ||
4070 | 432 | − 28.27 | 26.30 | 14 | 381 | ||
4 | 234 | 12 | 16 | − 27.95 | 26.12 | 1 | 313 |
850 | 48 | − 28.08 | 26.20 | 4 | 324 | ||
1850 | 108 | − 28.08 | 26.20 | 15 | 300 | ||
3234 | 192 | − 28.10 | 26.21 | 26 | 291 | ||
5 | 352 | 12 | 19 | − 28.10 | 26.22 | 2 | 336 |
1302 | 48 | − 28.09 | 26.20 | 18 | 322 | ||
2852 | 108 | − 28.11 | 26.22 | 35 | 320 | ||
5002 | 192 | − 28.12 | 26.22 | 119 | 300 |
Convergence analysis regarding displacements for the BBC model
ORD | NDOF | NE | NIP | u _{1} | u _{2} | PT (s) | TNI |
---|---|---|---|---|---|---|---|
1 | 24 | 12 | 1 | − 10.75 | 20.45 | 1 | 300 |
70 | 48 | − 19.75 | 25.37 | 1 | 300 | ||
234 | 192 | − 25.24 | 27.35 | 1 | 300 | ||
850 | 768 | − 27.90 | 28.20 | 1 | 270 | ||
1850 | 1728 | − 28.58 | 28.40 | 2 | 252 | ||
3234 | 3072 | − 28.85 | 28.49 | 4 | 241 | ||
5002 | 4800 | − 28.98 | 28.53 | 6 | 233 | ||
7154 | 6912 | − 29.06 | 28.56 | 8 | 230 | ||
2 | 70 | 12 | 13 | − 25.13 | 27.61 | 1 | 300 |
234 | 48 | − 28.53 | 28.36 | 1 | 300 | ||
494 | 108 | − 29.02 | 28.52 | 1 | 300 | ||
850 | 192 | − 29.13 | 28.57 | 2 | 275 | ||
1302 | 300 | − 29.18 | 28.59 | 3 | 262 | ||
1850 | 432 | − 29.20 | 28.60 | 4 | 254 | ||
3 | 140 | 12 | 16 | − 28.58 | 28.20 | 1 | 339 |
494 | 48 | − 29.17 | 28.57 | 1 | 336 | ||
1064 | 108 | − 29.21 | 28.60 | 2 | 347 | ||
1850 | 192 | − 29.23 | 28.62 | 4 | 353 | ||
2852 | 300 | − 29.24 | 28.63 | 8 | 359 | ||
4070 | 432 | − 29.25 | 28.63 | 14 | 381 | ||
4 | 234 | 12 | 19 | − 29.13 | 28.54 | 1 | 300 |
850 | 48 | − 29.21 | 28.61 | 5 | 279 | ||
1850 | 108 | − 29.23 | 28.62 | 12 | 264 | ||
3234 | 192 | − 29.24 | 28.63 | 13 | 244 | ||
5 | 352 | 12 | 46 | − 29.18 | 28.58 | 4 | 300 |
1302 | 48 | − 29.23 | 28.62 | 14 | 265 | ||
2852 | 108 | − 29.24 | 28.63 | 32 | 245 | ||
5002 | 192 | − 29.24 | 28.63 | 46 | 235 |
Partially loaded membrane
Convergence analysis for displacements and equivalent stresses
ORD | NDOF | NE | PT (s) | TNI | u_{2} (A) | u_{2} (B) | σ_{eq} (C) | σ_{eq} (D) |
---|---|---|---|---|---|---|---|---|
1 | 32 | 18 | 1 | 300 | − 8.45 | − 6.51 | 284.59 | 124.25 |
98 | 72 | 1 | 300 | − 8.77 | − 6.55 | 255.39 | 99.85 | |
100 | 162 | 1 | 200 | − 8.86 | − 6.55 | 254.53 | 88.46 | |
338 | 288 | 1 | 260 | − 8.89 | − 6.56 | 258.60 | 82.48 | |
1250 | 1152 | 2 | 214 | − 8.86 | − 6.59 | 273.17 | 75.65 | |
2738 | 2592 | 4 | 207 | − 8.81 | − 6.61 | 283.27 | 76.41 | |
4802 | 4608 | 8 | 200 | − 8.76 | − 6.62 | 290.58 | 77.60 | |
7442 | 7200 | 13 | 200 | − 8.73 | − 6.62 | 296.02 | 78.40 | |
2 | 98 | 18 | 1 | 300 | − 8.91 | − 6.77 | 272.72 | 116.04 |
338 | 72 | 1 | 300 | − 8.68 | − 6.78 | 300.23 | 58.83 | |
722 | 162 | 3 | 300 | − 8.67 | − 6.76 | 304.76 | 64.61 | |
1250 | 288 | 3 | 219 | − 8.67 | − 6.74 | 311.30 | 70.87 | |
1922 | 450 | 6 | 213 | − 8.67 | − 6.73 | 314.77 | 74.61 | |
3 | 200 | 18 | 2 | 300 | − 8.66 | − 6.76 | 311.55 | 57.91 |
722 | 72 | 3 | 300 | − 8.67 | − 6.74 | 324.35 | 73.45 | |
1568 | 162 | 4 | 219 | − 8.67 | − 6.71 | 322.23 | 77.67 | |
2738 | 288 | 6 | 209 | − 8.67 | − 6.70 | 322.07 | 78.57 | |
4 | 338 | 18 | 5 | 300 | − 8.66 | − 6.72 | 317.27 | 40.11 |
1250 | 72 | 7 | 258 | − 8.67 | − 6.69 | 328.85 | 78.95 | |
2738 | 162 | 12 | 209 | − 8.67 | − 6.67 | 328.49 | 79.10 |
Considering the equivalent stress levels at points C and D, the rate of convergence is improved by increasing the element order. Except for the two less refined linear and quadratic meshes (18 and 72 elements), the equivalent stresses always converge when the number of elements is increased. In summary, the most appropriate orders for the present example, in the context of Table 3, are probably the cubic and fourth degrees, because of the stability of values regarding displacements and stresses. As in the first example, the processing times and number of iterations required are small, even for the most refined discretizations.
Rubber sealing
Convergence analysis regarding applied forces and equivalent stresses
ORD | NDOF | NE | PT (s) | TNI | p _{2} | σ_{eq} (A) | σ_{eq} (B) | σ_{eq} (C) |
---|---|---|---|---|---|---|---|---|
1 | 36 | 20 | 1 | 400 | − 312.49 | 363.76 | 398.85 | 273.26 |
110 | 80 | 1 | 400 | − 231.06 | 346.52 | 356.53 | 295.32 | |
378 | 320 | 1 | 500 | − 184.22 | 341.75 | 318.72 | 250.30 | |
1394 | 1280 | 3 | 500 | − 161.40 | 325.72 | 285.33 | 223.60 | |
3050 | 2880 | 8 | 544 | − 153.41 | 312.68 | 268.52 | 216.86 | |
5346 | 5120 | 18 | 600 | − 149.93 | 304.71 | 259.73 | 214.02 | |
8282 | 8000 | 35 | 655 | − 147.12 | 299.90 | 255.21 | 212.63 | |
2 | 110 | 20 | 1 | 400 | − 195.09 | 360.35 | 380.08 | 291.69 |
378 | 80 | 3 | 500 | − 157.33 | 330.65 | 270.75 | 221.22 | |
806 | 180 | 5 | 500 | − 148.48 | 310.04 | 248.03 | 212.46 | |
1394 | 320 | 8 | 500 | − 145.00 | 298.48 | 245.40 | 212.15 | |
2142 | 500 | 14 | 558 | − 142.91 | 292.79 | 246.84 | 212.66 | |
3050 | 729 | 21 | 600 | − 142.23 | 289.98 | 248.04 | 213.05 | |
5346 | 1280 | 43 | 661 | − 141.78 | 287.88 | 248.99 | 213.51 | |
3 | 224 | 20 | 1 | 500 | − 158.68 | 338.28 | 343.00 | 252.12 |
806 | 80 | 3 | 500 | − 144.46 | 303.53 | 244.44 | 204.27 | |
1748 | 180 | 11 | 600 | − 141.82 | 289.25 | 243.49 | 206.56 | |
3050 | 320 | 20 | 638 | − 141.23 | 286.45 | 246.65 | 209.14 | |
4712 | 500 | 37 | 740 | − 141.26 | 285.71 | 247.69 | 210.60 | |
4 | 378 | 20 | 6 | 500 | − 148.24 | 308.08 | 343.76 | 243.04 |
1394 | 80 | 14 | 600 | − 141.55 | 297.17 | 244.37 | 200.44 | |
3050 | 180 | 42 | 700 | − 141.09 | 288.39 | 243.94 | 204.44 | |
5 | 572 | 20 | 23 | 530 | − 142.71 | 289.88 | 332.96 | 245.53 |
2142 | 80 | 40 | 500 | − 141.28 | 298.70 | 244.69 | 200.73 | |
4712 | 180 | 117 | 645 | − 140.66 | 288.23 | 243.77 | 204.30 |
Conclusions
A finite-element formulation to solve structural problems involving hyperelastic materials under plane-stress conditions has been developed. The elements are isoparametric triangular membranes of any order based on positional description. Lagrangian strain and stress measures from nonlinear solid mechanics are adopted: deformation gradient, right Cauchy–Green stretch tensor (and its invariants) and the second Piola–Kirchhoff stress tensor. Two hyperelastic models, usually employed for elastomers, have been selected: the compressible neo-Hookean model (nH); and the general incompressible Rivlin–Saunders model (RS). The 3D constitutive relations are condensed according to the plane-stress conditions and the resultant 2D compact forms for both models are provided. This condensation procedure requires the solution of nonlinear equations. It has been demonstrated that, in the case of the nH model, the expression to determine the out-of-plane normal strain is not explicit and, thus, a simple Newton algorithm is employed to obtain this strain component from the plane strains. The formulas to determine the consistent tangent operator, which is a fourth-order tensor required in the present Newton–Raphson method, are provided in the appendices considering the 2D constitutive forms.
Three structural problems involving hyperelastic materials under plane-stress conditions are analyzed to assess the performance of the present finite-element approach: the Cook’s membrane, a bending-dominated problem with a singularity point; the partially loaded membrane, which presents a high level of compression together with severe mesh distortion; and the rubber sealing, a more realistic engineering problem. These benchmark problems have been employed to examine the convergence rate regarding displacements, applied forces, and stresses. Since the degree of approximation is generic, several meshes with different numbers of elements and orders (up to fifth degree) have been used to analyze each problem with the same computer code developed. As expected, mesh refinement provides more accurate results and the convergence rate obtained with the higher orders is similar to that presented by some mixed formulations from the scientific literature. The main advantage of the present method in comparison with these alternative formulations is the mathematical simplicity, i.e., no special techniques and no complex mathematical formulations are needed. Results also indicate that the finite-element formulation proposed here can predict the correct structural behavior in both incompressible and near-incompressible regimes. Therefore, in the context of the present paper, general locking problems are avoided by mesh refinement. These locking phenomena are usually associated with the incompressibility assumption, stress concentration, mesh distortion, complex geometries, and bending-dominated problems. It should also be highlighted that the so-called hourglass instabilities, which are common in mixed formulations, have not been observed in the present analyses. In short, the finite-element formulation proposed here is showed to be simple, robust, accurate, and reliable. Thus, it can be extended to other types of numerical analyses with highly deformable materials under plane-stress conditions as, for example, elastoplasticity, viscoelasticity, thermoelasticity and anisotropy.
Some contributions of the present study can be cited: convergence regarding displacements and stresses; use of higher orders; and development of the consistent tangent operator for the hyperelastic models nH and RS in plane-stress conditions. Although it is well known that mesh refinement leads to more accurate results, the influence of the mesh on the stress field and the use of higher orders of approximation are rarely seen. Moreover, some details of the consistent tangent operator provided here are usually omitted.
Notes
Acknowledgements
The author thanks all the support provided by the Materials Engineering Department (Lorena School of Engineering, University of São Paulo), as well as the Structural Engineering Department (São Carlos School of Engineering, University of São Paulo) for granting remote access to their cluster.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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