Orthotropic Hill Plasticity Model

In the field of structural mechanics, the plastic deformation processes of many materials are anisotropic in nature. Examples include composite materials, titanium alloys, additively manufactured structures, and multiscale materials. In these cases, traditional isotropic yield criteria are no longer sufficient to accurately describe the behavior, and appropriate models are required to characterize anisotropic plastic yielding. Commonly used anisotropic yield functions include the Hill, Barlat, Banabic, and Cazacu models. Among them, the Hill model was proposed relatively early and has a wide range of applications. It performs particularly well in describing orthotropic materials and is extensively used in structural finite element analysis. hill_augmented_manufacturing The Hill model is named after Professor R. Hill, who was appointed Professor of Applied Mathematics at the University of Nottingham in 1953. His 1950 work “The Mathematical Theory of Plasticity” laid the foundation for modern plasticity theory. He is widely regarded as one of the most significant contributors to solid mechanics in the second half of the 20th century. It’s important to note that the Hill model is not the only yield criterion for orthotropic anisotropic materials. Alternatives include Barlat models with 3 or 6 parameters, or modified versions of the Hill yield criterion. This article focuses only on the classical Hill model, but similar approaches can be used to determine parameters for other yield models. The Hill model is applicable for analyzing plastic deformation in anisotropic materials and can be viewed as a generalized form of the von Mises yield criterion tailored for anisotropic behavior. It is particularly useful for orthotropic plastic materials in structural engineering applications. The Hill yield criterion is given by: hill_yield_equation For shell elements, the yield function is given by: hill_yield_equation2 Where σ represents the stress components. For 3D models, F to H are six anisotropy calibration coefficients, also known as Hill parameters, which can be determined through material experiments. For shell elements, only four parameters F to N are required. In computational mechanics, solid elements typically use yield stress ratios (or yield stresses) R11, R22, R33, R12, R13, R23 to calculate Hill parameters. To determine the yield stress ratios, the material’s yield stresses under different loading conditions must be measured: σ11, σ22, σ33 are obtained from tensile tests. σ12, σ13, σ23 are obtained from shear tests. For shell elements, Lankford parameters are used to determine the Hill parameters. A Lankford parameter rₐ is defined as the ratio of plastic strain in the plane to that in the thickness direction. For orthotropic anisotropy, rₐ can be obtained from simple tensile tests: r₀₀: tensile direction aligned with the orthotropic axis 1. r₉₀: tensile direction perpendicular to the orthotropic axis 1. Once r₀₀, r₄₅, r₉₀ are measured, the Hill parameters can be calculated. Higher Lankford values indicate better formability. Most finite element analysis software requires the user to input either R values or Lankford parameters for the Hill model. Once these are specified, the Hill yield criterion equation can be fully determined. Support for the orthotropic Hill model in WelSim The complexity of the Hill model lies in parameter input and solving the orthotropic nonlinear behavior. The general-purpose simulation software WESLIM supports transient analysis involving the Hill model. The free material editing software MatEditor supports Hill material model input. Currently, MatEditor supports three types of orthotropic Hill plastic material models and can generate OpenRadioss material input scripts: Law32, Law73/74, and Law93. mateditor_hill_orthotropic Conclusion The Hill model effectively describes orthotropic yield behavior, especially in tension-compression directions and in scenarios with minimal shear deformation. It can also be used to model the mechanical behavior of metallic lattice structures in additive manufacturing. Hill parameters can be determined via R or Lankford parameters. When these parameters are unavailable from experiments, they can be approximated using curve fitting methods. WelSim already provides support for the Hill model and can be integrated with OpenRadioss for transient simulations involving orthotorpic plastic deformation. WelSim and authors have no direct affiliation with developers of OpenRadioss. References to OpenRadioss here are solely for technical blogging and software usage discussion.

正交各项异性Hill 塑性模型

在结构力学领域，很多材料的塑性变形过程是各项异性的，如复合材料，钛合金，增材制造结构，多尺度材料等。这时传统的各项同性屈服准则就无法准确描述此过程，需要有合适的模型来描述各项异性塑性屈服，常用的各项异性屈服函数有：Hill 系列、Barlat 系列、Banabic和 Cazacu等。其中，Hill模型是提出较早，且应用范围广。尤其是在描述正交各向异性材料，Hill模型有着很好的表现，被广泛应用于结构有限元计算中。 Image Hill塑性模型是以R. Hill教授命名的，他在1953年被任命为诺丁汉大学应用数学教授。于1950年发表的《塑性数学理论》奠定了塑性理论的基础。被广泛认为是20世纪下半叶固体力学基础最重要的贡献者之一。 Hill模型并不是描述正交各项异性屈服的唯一准则，例如3参数或者6参数的Barlat本构模型，或者修改版的Hill屈服准则。本文仅对经典的Hill模型进行讨论，其他屈服模型也可以采用类似的方法进行确定。 Hill适用于各项异性的塑性变形分析。可以看作是用于各项异性屈服行为的von Mises屈服准则的通用形式。在实际的结构工程中，常用于正交各项异性的塑性材料。 Hill 的屈服准则如下： Image 对于壳单元，屈服函数如下 Image sigma是应力分量，对于三维模型，F-H为六个各向异性校准系数，又称作Hill参数，可以通过材料实验得到。对于壳单元，只需要F-N 4个参数。 在计算上，固体单元使用 屈服应力率 （或屈服应力）R11, R22, R33, R12, R13, R23来计算Hill参数。为了得到屈服应力率，材料在两种载荷工况下的屈服应力需要被测量。其中，屈服应力sigma11, sigma22, sigma33从拉伸测试中获得。屈服剪应力 sigma12, sigma13, sigma23从剪切测试中获得。 对于壳单元，会使用Lankford参数组来决定Hill参数。Lankford参数r_a是 平面上与厚度方向上塑性应变的比值。对于正交各项异性，r_a可以从简单的拉伸测试中获得。比如r00 从拉伸测试中，载荷方向与正交第一方向一致。r90的载荷方向与正交第一方向垂直。一旦Landford参数r00, r45, r90获得，便可以计算得到Hill参数。Landford数值越大，表示材料有更好的延展性。 在大多数有限元分析软件中，Hill模型要求用户输入R或者Landford参数。通过确定R或者Landford参数，即可确定Hill屈服准则方程。 WelSim对正交各项异性Hill模型的支持 Hill模型的复杂性在于模型参数的输入，和正交各项异性的非线性的求解。通用仿真软件WELSIM支持含有Hill的瞬态动力学计算，免费的材料编辑软件MatEditor支持Hill材料模型的输入。目前MatEditor已经支持了三种正交各项异性Hill塑性材料模型，并支持生成OpenRadioss的材料卡片Law32， Law73/74和Law93。 Image 总结 Hill模型可以描述材料的正交各项异性屈服行为。尤其是在拉压方向，以及剪切变形较小的场合，Hill模型也可以描述金属增材点阵结构的力学行为。 Hill参数可以通过R或者Landford参数确定，当这些参数无法从实验中获得时，也可以通过曲线拟合的方法近似得到。 WelSim已经对Hill模型有了一定支持，能够联合OpenRadioss进行含有各向异性塑性变形的瞬态动力学计算。 WelSim与作者和OpenRadioss开发者没有直接关系。这里引用OpenRadioss仅用作技术博客文章与软件使用的参考。