Linear Elastic Materialsmi@se.edu.cn
Linear Elastic Materials
Outline·Introduction（引言）·Linear elasticmodel（线弹性本构）·Matrixrepresentation（线性本构的矩阵表示）·Symmetryof stiffness/compliancetensor（刚度/柔度张量的对称性）·Linearelasticanisotropic models（线性各向异性模型）·Linearelasticorthotropic model（线性正交对称模型）·Linearelasticcubicmodel（线性立方对称模型）·Linearelastic isotropicmodel（线性各向同性本构）·Smallstretchesbutlargerotations（小应变大转动）2
Outline • Introduction（引言） • Linear elastic model（线弹性本构） • Matrix representation（线性本构的矩阵表示） • Symmetry of stiffness/compliance tensor（刚度/柔度张量 的对称性） • Linear elastic anisotropic models（线性各向异性模型） • Linear elastic orthotropic model（线性正交对称模型） • Linear elastic cubic model（线性立方对称模型） • Linear elastic isotropic model（线性各向同性本构） • Small stretches but large rotations（小应变大转动） 2
Introduction: Relations that characterize the mechanical behavior ofmaterials. Perhaps one of the most challenging fields in mechanics,due to the endless variety of materials and loadings The mechanical behavior of solids is normally defined byconstitutive stress-strain relationso= f(c,c,t,T,..)Linear elastic model (Hooke's law)Elastic-plastic modelVisco-elastic modelVisco-plastic model3
Introduction • Relations that characterize the mechanical behavior of materials • Perhaps one of the most challenging fields in mechanics, due to the endless variety of materials and loadings • The mechanical behavior of solids is normally defined by constitutive stress-strain relations 3 • Linear elastic model (Hooke’s law) • Elastic-plastic model • Visco-elastic model • Visco-plastic model σ f t ε, , , , ε T
Introduction: Neglect strain rate, time and loading history dependency: Set aside thermal, electrical, pore-pressure, and otherloadsInclude only mechanical loadsAssume linear stress-strain relationship Defined as materials that recover original configurationwhen mechanical loads are removedAgree well with experimental tests of metals0,=ESteelCast IronAluminumA
• Neglect strain rate, time and loading history dependency • Set aside thermal, electrical, pore-pressure, and other loads • Include only mechanical loads • Assume linear stress-strain relationship • Defined as materials that recover original configuration when mechanical loads are removed • Agree well with experimental tests of metals Introduction x x E 4
Linear Elastic Model. Hooke's law in 1D:C=E. In 3D, one might generalize this in tensor form as=C:& O, =Cjku;&=S: j=SijkOl Most generally, C has 81 independent components: Thanks to the (minor) symmetry propertiesO,=Oj =Cjh=G = =Cjk =jikl,Ai: The number of independent components is reduced to 36.5
• Hooke’s law in 1D: Linear Elastic Model • In 3D, one might generalize this in tensor form as : ; : σ C ε ij ijkl kl ij ijkl kl C S ε S σ • Most generally, C has 81 independent components. • Thanks to the (minor) symmetry properties , ij ji kl kl kl lk ij ij C C C C ij ji kl lk • The number of independent components is reduced to 36. 5