Mechanics theories for composite materials
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摘要: 各向同性材料的力学理论已基本完善，但各向异性复合材料的力学理论除线弹性外，皆未成熟，尤其破坏和强度分析，依然还是固体力学面临的一个最大挑战。在过去25年中，为推动复合材料力学学科发展，作者创建了一系列解析理论，包括任意连续纤维、短纤维及颗粒增强复合材料的本构与内应力计算理论-桥联模型，将基体均值应力转换到真实值的真实应力理论，基于物理原理建立的基体破坏准则，预报任意层合结构层间开裂或分层的层间基体应力修正法及超弹性材料的增量型本构理论。基于这些理论，几乎所有两相复合材料的破坏问题，皆有望通过解析公式获得有效解决，只要该复合材料的孔隙率可忽略。其中，桥联模型已得到国内外同行广泛认可，他人应用该理论公开发表的研究论文，已超过250篇。基体真实应力理论，尽管前不久才建立起来，他人应用也已达20篇。本文简要介绍了作者建立的这些理论及如何据此解决众多复合材料挑战性问题。Abstract: Whereas mechanics theories for isotropic materials have been nearly matured, essentially only the linear elasticity theories for anisotropic composite materials are well established. All of the other mechanical behaviors of the composites are not well understood. Specifically, the failure and strength analysis for the composites still remains to be one of the greatest challenges in solid mechanics. During the last 25 years, in order to advance the development in the mechanics of composite materials, this author has established a series of analytical theories. They include the constitutive and internal stress calculation theory, named Bridging Model, for composites reinforced with continuous or short fibers or particles, the true stress theory for converting a homogenized stress of the matrix in a composite into a true value, the failure criteria for matrix failures established on a physics based principle, the interlaminar matrix stress modification method for predicting interlaminar fracture or delamination of any laminated structure, and the incremental constitutive relation for hyperelastic materials. Based on these theories, almost all of the failure problems of two-phase composites can be efficiently resolved through analytical formulae, as long as void contents in the composites can be neglected. Amongst, Bridging Model has been known world-widely, and more than 250 publications have been made by people other than the author’s group using Bridging Model as a theoretical tool. Furthermore, the others’ publications based on the matrix true stress theory have reached a number of 20, although this theory has been established only recently by the author. A brief summary on the establishment of the author’s theories and how to apply them to resolve challenging problems in composites and their structures is presented in the paper.
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图 1 连续纤维 (a) 和短纤维复合材料 (b) 的代表性单元(RVE)(l/a≈1对应颗粒复合材料)
Figure 1. Representative unit (RVE) of a continuous fiber composite (a) and a short fiber composite (b) (l/a≈1 corresponds to a particle composite)
a—Fiber radius; b—Matrix radius; L—Half length of matrix; l—Half length of fiber; Vf—Fiber volume fraction; (r, q, z)—Cylindrical coordinates; (x1, x2, x3)—Rectandular coordinates; Ω1—A portion of RVE with fiber; Ω2—A portion of RVE without fiber
图 14 复合材料结构破坏和强度分析流程图
Figure 14. A flow chart to show failure analysis and strength predictions for any composite structure
图 2 超弹性本构模型预报与Treloar的橡胶实验[73]对比：(a) 模量-名义应变(E-εe)曲线；(b) 泊松比-名义应变(ν-εe)曲线；((c)~(e)) 单轴拉伸实验拟合模型参数后预报的单轴拉伸、等值双轴拉伸、纯剪切直到破坏的曲线与实验对比；((f)~(h)) 等值双轴拉伸实验拟合模型参数后预报的单轴拉伸、等值双轴拉伸、纯剪切直到破坏的曲线与实验对比；((i)~(k)) 纯剪切实验拟合模型参数后预报的单轴拉伸、等值双轴拉伸、纯剪切直到破坏的曲线与实验对比
Figure 2. Hyperelastic theories verified through Treloar’s test data on a rubber[73]: (a) Modulus versus equivalent strain (E-εe) curves; (b) Poisson’s ratio versus equivalent strain (ν-εe) curves; ((c)-(e)) Comparisons between the measured and predicted uniaxial tension, equalbiaxial tension and pure shear curves by different models fitted with uniaxial tension test data; ((f)-(h)) Comparisons between the measured and predicted uniaxial tension, equalbiaxial tension, pure shear curves by different models fitted with equalbiaxial tension test data; ((i)-(k)) Comparisons between the measured and predicted uniaxial tension, equalbiaxial tension, pure shear curves by different models fitted with pure shear test data
图 3 对应不同标度应力的复合材料性能示意图
Figure 3. Schematic disgram to show composite properties at different scales of stresses
σ—Stress; ε—Strain; E—Elastic modulus; ET—Hardening modulus
图 4 RVE界面开裂后纤维和基体之间相对滑移示意图：(a) 纵截面；(b) 横界面
Figure 4. Schematic of relative slippage displacement between debonded fiber and matrix interfaces in a RVE: (a) Longitudinal cross-section; (b) Transverse cross-section
D, D', F—Label of the point; σ12 0—In-plane shear stress applied on the RVE; d—Relatic slippage displacement between debonded fiber and matrix interfaces; Ψ—Half debonding angle
图 5 机织纤维 (a)、针织纤维 (b)、短纤维增强复合材料 (c) 重复单胞示意图
Figure 5. A repeating unit cell for woven fabric composite (a), knitted fabric composite (b), short fiber reinforced composite (c)
图 7 层合板的双悬臂梁(DCB) (a) 和端部开口弯曲(ENF) (b) 实验和有限元模拟
Figure 7. Experimental and finite element approaches on double cantilever beam (DCB) (a) and end-notched flexure (ENF) (b) samples
H—Thickness of the sample; PI, PII—Peak load of DCB, ENF ; δI, δII—Corresponding displacement; a0—Length of an initial crack
图 8 (a) 轴向拉伸引起的纤维撕裂破坏；(b) 轴向压缩引起的纤维偏折破坏[91]；(c) 轴向加载在纤维偏折坐标系的应力分量
Figure 8. (a) Fiber splitting failure under a longitudinal tension; (b) Fiber kinking failure under a longitudinal compression[91]; (c) A longitudinal load induced stress components in the misaligned coordinate system
σL—Axial stress; τ—Shear stress; $\theta_0^{{\rm{f}},0} $—Initial fiber misalignment angle; σT—Transverse stress; $x_1^{\rm{I}} $, $x_2^{\rm{I}} $—Misaligned coordinate system
图 9 (a) 基体斜截面上拉伸正应力对应的破坏应力状态；(b) 拉伸破坏面包络线由基体拉伸和剪切强度构建的抛物线近似
Figure 9. (a) Failure stress state of an inclined cross-section with a positive outward normal stress; (b) Tensile failure envelope approaximated by a parabola determined from matrix’s tensile and shear strengths
图 10 单向复合材料受横向压缩破坏图
Figure 10. Failure disgram of a unidirectional composite under a transverse compression
x2—Load direction; x3—Fiber direction
图 11 基体压缩引起的复合材料破坏示意图
Figure 11. Schematic of a matrix compression induced composite failure
{σ11, σ22,...}—Stresses on a unidirectional composite composite; {σn, τnt, τns}—Normal and two shear stresses of the composite on an inclined cross-section; {$\tau _{\rm nt}^{\rm{m}} $, $\tau _{\rm n}^{\rm{m}} $, $\tau _{\rm ns}^{\rm{m}} $}—Normal and two shear stresses of the matrix on the inclined cross-section
图 12 复合材料弱奇异和应力集中截面邻域示意图(t为主层厚)
Figure 12. Schematic neibourhoods of weak singularity and stress concentration points in a composite (t is the thickness of a primary layer)
图 13 任意纤维(编织纤维结构为例)复合材料结构分析示意图
Figure 13. Schematic diagram to shown structural analysis for any fibrous composite structure (Using braided fabric composite as an examble)
表 1 复合材料破坏分类
Table 1. Classification for composite failures
Mode Characteristics Fatal failure Fiber tensile failure in a primary layer Fiber compressive failure in a primary layer Fiber splitting or kinking failure in a primary layer Non-fatal failure Matrix tensile failure in a primary layer Matrix compressive failure in a primary layer Damage type failure Interface crack between fiber and matrix Delamination between adjacent primary layers Non-fatal failure in a laminated composite Strength type failure There is a fatal failure A critical strain is attained after a non-fatal failure Ultimate failure of composite structure A strength type failure occurs outside neighborhoods of stress singularity (constrained boundary, loading point, etc.) and week singularity (hole edge, free edge, tapered laminate, etc.) 
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表 2 复合材料结构破坏与强度分析所需材料性能数据
Table 2. Material property parameters needed for failure and strength analysis of a composite structure
Property Source Remark Fiber elastic constants Material data sheet/fiber supplier/tests on UD Each fiber must provide 5 elastic constants (for transversely isotropic). Can be retrieved from Eqs. (12). Fiber tensile & compressive strengths Tests on UD/fiber supplier/material data sheet Each fiber must provide 2 strengths. Can be retrieved from Eqs. (55.1) and (55.2). Matrix elastic constants Tests on casting samples of matrix/matrix supplier Each matrix must provide its Young’s modulus and Poisson’s ratio Matrix tensile & shear strengths Tests on casting samples of matrix Each matrix must provide its measured tensile and shear strengths, with shear better obtained on Iosipescu method Matrix compressive strength Transverse compressive strength of UD/test on matrix/matrix supplier Each matrix must provide its compressive strength, best retrieved from Eq. (55.3) with $ K_{22}^c $ determined by Eqs. (56)[50,97] Matrix tensile curve Test on casting sample of matrix Define matrix tensile yield strength and hardening moduli, as long as the piece linear segments are near to the curve Matrix compress. curve Test on casting sample of matrix Define matrix compressive yield strength and hardening moduli, as long as piece linear segments near to the curve UD transverse tensile strength Test on UD Each composite system must provide its transverse tensile strength to understand interface bonding status Critical displacements
δI and δIIDCB and ENF tests Preferably each pair of two different materials used in the structure should provide the DCB and ENF test data Fiber initial misalignment $\theta_c^{f,0} $=1.50 Empirical data Neighborhood ranges n1=4, n2=24 Empirical data Note: UD—Unidirectional composite.
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