**1. Introduction**

Bending is frequently used as a test for determining various material properties. A four-point bend apparatus was used in [1] for getting stress–strain curves in the tension and compression of three materials: beryllium, cast iron, and copper. No significant difference between these curves and the corresponding curves from the conventional uniaxial tension and compression tests was found. Papers [2,3] applied the same method for determining the stress–strain curves for annealed copper and cement-based composites, respectively. The theoretical solution proposed in [1] was modified in [4]. Then, the method was used for determining the stress–strain curves for pure magnesium and S45C steel. Paper [5] studied the elastic/plastic behavior and the failure of CLARE laminates in bending experimentally. An elastic/plastic material model was proposed using these experimental data. Lightweight-aggregate concrete beams were tested in bending to failure in [6]. This work emphasized the location of the neutral axis at failure. Adhesively bonded bending specimens were employed in [7] for determining the bilinear elastic/plastic shear stress–strain behavior of acrylic adhesives. Hybrid sandwich structures were tested in bending in [8] to construct failure mode maps.

Experimental data should be usually supplemented with theoretical solutions for identifying material models. Most semi-analytic solutions have dealt with strain hardening materials [9–12]. Paper [12] also accounted for plastic anisotropy and tension/compression asymmetry. The representation of strain or work hardening in the strain reversal region has been simplified in these works. Paper [13] developed a semi-analytic method to treat this region without any simplification. This method was combined with various constitutive equations to describe pure bending and bending under tension of wide sheets [14]. Under certain conditions, viscoplastic or strain-hardening viscoplastic constitutive equations are required to analyze bending processes adequately [15–17]. Many material models of this type are available in the literature [18–24]. For identifying constitutive equations, it is desirable to have a theoretical solution for a general material model. The motivation of

**Citation:** Alexandrov, S.; Lyamina, E. Analysis of Strain-Hardening Viscoplastic Wide Sheets Subject to Bending under Tension. *Metals* **2022**, *12*, 118. https://doi.org/10.3390/ met12010118

Academic Editor: Gabriel Centeno Báez

Received: 17 December 2021 Accepted: 4 January 2022 Published: 7 January 2022

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the present paper is to provide a semi-analytic solution of the same level of complexity as available solutions but for quite a general material model. It is believed that such a solution is useful for identifying constitutive equations. The solution is based on the general method proposed in [13].

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The solution found is facilitated by using the equivalent strain instead of the space coordinate as an independent variable. Similar changes of independent variables have been successfully used in several works [25–27]. In particular, the process of bending was analyzed in [26]. However, in all these cases, it has been assumed that the yield stress depends on the single kinematic quantity, the equivalent plastic strain. The novelty of the present solution is that the yield stress is an arbitrary function of the equivalent strain and the equivalent strain rate. An important property of the new solution is that process parameters depend on the loading speed. In addition, the solution shows that some assumptions used in simplified formulations are not valid and corresponding numerical results may lead to the misinterpretation of experimental data. ‐ ‐ 
