东南大学：《固体力学基础》课程教学课件（英文讲稿）05 Hypo-elastic Materials

Hypo-elastic Materialsmi@se.ed.cn

Hypo-elastic Materials

Outline·Introduction（弓言）·Nonlinearelastic shearmodel（非线性弹性剪切模型）·Calibration of nonlinear elastic shear model（模型校准2

Outline • Introduction（引言） • Nonlinear elastic shear model（非线性弹性剪切模型） • Calibration of nonlinear elastic shear model（模型校准） 2

Lntroduction福? Hypoelasticity is used to modelmaterials that exhibit nonlinear, butStrainenergyreversible, stress-strain behavior evendensityCat small strains The specimen deforms reversibly: ifyou remove the loads, the solidreturns to its original shape.: The strain in the specimen depends only on the stressapplied to it; it does not depend on the rate of loading orthe history of loading. We will assume here that the material is isotropic3

Introduction 3 • The strain in the specimen depends only on the stress applied to it; it does not depend on the rate of loading or the history of loading. • We will assume here that the material is isotropic. • Hypoelasticity is used to model materials that exhibit nonlinear, but reversible, stress-strain behavior even at small strains. • The specimen deforms reversibly: if you remove the loads, the solid returns to its original shape

Introduction. Strains and rotations are assumed to be small. We useinfinitesimal strain and Cauchy stress.? Existence of a strain energy density guarantees thatdeformations of the material are perfectly reversible. If the material is isotropic, the strain energy can only be afunction of invariants of the strain tensor, i.e. threeprincipal strains.? It is usually more convenient to use the three fundamentalscalar invariants:I,==+82+83,n, =nn;(,8,-8,))=82 +8283 +8381det[8, -8,8, = 01= det-S.8iGu6,6,838,3+,c,2-1,8,+1,=064

• Strains and rotations are assumed to be small. We use infinitesimal strain and Cauchy stress. • Existence of a strain energy density guarantees that deformations of the material are perfectly reversible. • If the material is isotropic, the strain energy can only be a function of invariants of the strain tensor, i.e. three principal strains. • It is usually more convenient to use the three fundamental scalar invariants: Introduction 4 3 2 1 2 3 det 0 0 ij j n i ij n ij n n n n n I I I                       1 1 2 3 2 1 2 2 3 3 1 3 1 2 3 1 2 1 det 6 kk ii jj ij ji ij ijk rst ir js kt I I I                                       

Nonlinear Elastic Shear Model: In most practical applications, nonlinear behavior is onlyobserved when the material is subjected to sheardeformation (characterized by I,), whereas stress varieslinearly with volume changes (characterized by I). Note the slightly different definition of I, which rendersus great simplification in deriving the inverse relation:al,1al=0,l=(sf,-1eu5n)uoI =6k =6.-306j061n+1)/2nau2no.6al.al0-KI?UI.KI=aI206jn+1ao06600/2OUC30605

• In most practical applications, nonlinear behavior is only observed when the material is subjected to shear deformation (characterized by I2 ), whereas stress varies linearly with volume changes (characterized by I1 ). • Note the slightly different definition of I2 , which renders us great simplification in deriving the inverse relation: Nonlinear Elastic Shear Model 5         1 2 1 2 2 0 0 0 2 1 2 1 2 2 1 2 1 1 2 2 0 0 0 1 1 2 2 0 2 2 0 0 1 1 1 , 2 3 1 2 , 2 1 1 3 3 n n n n kk ij ij ij kk ll ij ij ij ij n n ij kk ij ij kk ij i ij kk i i j j j n I I I I U U I I KI K I I I I I I n K                                                                                            

Nonlinear Elastic Shear Model? Strain in terms of stress16%2dC,=K=k=3K8=8k8kk2kkA1.33K60Sn1-1)/2n1%0813a.160.0UO1钻kki19K3oO=3K=0 k-2g0..0+208O.LKA3-20:(0,0-50u0m)1,=g-C一6

• Strain in terms of stress Nonlinear Elastic Shear Model 6         2 0 2 2 0 1 1 2 2 2 0 0 0 1 2 2 2 0 0 0 0 1 1 3 3 3 1 3 1 1 9 3 n ij kk ij kk ij kk kk kk n ij n n n kk ij kk ij ij kk ij ij kk ij ij kk ij I K K K K K I I                                                                                 1 2 2 2 2 0 0 2 1 2 1 2 2 2 2 0 2 2 0 0 0 2 0 1 3 2 2 3 1 1 , 2 3 1 1 2 3 n ij ij kk ll kk ll ij ij kk l n ij ij kk ll l n I K I I I I                                                              

Calibration of Nonlinear Elastic Shear ModelOo..019K6Hydrostatic stress statepCi1 = 22 = O33 = p = 8u = 822 二839K3Ka. Pure shear1CO12=021 = T=812=89K3O.Uniaxial tensionO9K9K0Ogu=g,O80)9KV39KJo3aa

• Hydrostatic stress state Calibration of Nonlinear Elastic Shear Model 7       0 11 22 33 11 22 1 2 2 2 0 33 0 1 1 3 3 9 3 n I p p p p K                                 3 p K  • Pure shear 12 21 1 2 2 1 12 1 9 kk K               2 12 1 2 0 2 0 0 1 3 n kk                 0 0 n                 • Uniaxial tension     1 2 0 0 11 11 11 2 0 0 0 11 1 2 0 0 22 33 22 22 2 0 0 2 2 2 2 2 2 0 1 1 2 9 3 9 3 3 , 1 1 9 3 9 3 1 1 2 3 3 3 3 3 n n n n K K I K K                                                                                                        1 2  0 2 2 2 0 0 1 1 1 1 , 9 3 2 3 n ij kk ij ij kk ij ij ij kk ll I I K                                       