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Elasticity and Viscoelasticity CHAPTER 2 Elasticity and Viscoelasticity CHAPTER 2.1 Introduction to Elasticity and Viscoelasticity JEAN LEMAITRE Universite! Paris 6, LMT-Cachan, 61 avenue du President! Wilson, 94235 Cachan Cedex, France For all solid materials there is a domain in stress space in which strains are reversible due to small relative movements of atoms. For many materials like metals, ceramics, concrete, wood and polymers, in a small range of strains, the hypotheses of isotropy and linearity are good enough for many engineering purposes. Then the classical Hooke’s law of elasticity applies. It can be de- rived from a quadratic form of the state potential, depending on two parameters characteristics of each material: the Young’s modulus E and the Poisson’s ratio n. 1 c * ¼ A s s ð1Þ 2r ijklðE;nÞ ij kl @c * 1 þ n n eij ¼ r ¼ sij À skkdij ð2Þ @sij E E Eandn are identified from tensile tests either in statics or dynamics. A great deal of accuracy is needed in the measurement of the longitudinal and transverse strains (de Æ10À6 in absolute value). When structural calculations are performed under the approximation of plane stress (thin sheets) or plane strain (thick sheets), it is convenient to write these conditions in the constitutive equation. Plane stress ðs33 ¼ s13 ¼ s23 ¼ 0Þ: 2 3 1 n 6 À 0 7 2 3 6 E E 72 3 6 7 e11 6 7 s11 6 7 6 1 76 7 4 e22 5 ¼ 6 0 74 s22 5 ð3Þ 6 E 7 6 7 e12 4 5 s12 1 þ n Sym E Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 71 72 Lemaitre Plane strain ðe ¼ e ¼ e ¼ 0Þ: 332 133 223 32 3 s l þ 2ml 0 e 6 11 7 6 76 11 7 4 s22 5 ¼ 4 l þ 2m 0 54 e22 5 s12 Sym 2m e12 8 > nE > l ¼ < ð1 þ nÞð1 À 2nÞ ð4Þ with > > E : m ¼ 2ð1 þ nÞ For orthotropic materials having three planes of symmetry, nine independent parameters are needed: three tension moduli E1; E2; E3 in the orthotropic directions, three shear moduli G12; G23; G31, and three contraction ratios n12; n23; n31. In the frame of orthotropy: 2 3 2 3 e11 2 3 s11 6 7 1 n12 n13 6 7 6 7 À À 0006 7 6 7 6 E E E 76 7 6 7 6 1 1 1 76 7 6 e 7 6 n 76 s22 7 6 22 7 6 1 23 76 7 6 7 6 À 00076 7 6 7 6 E2 E2 76 7 6 7 6 76 7 6 e 7 6 1 76 s 7 6 33 7 6 00076 33 7 6 7 6 E3 76 7 6 7 ¼ 6 76 7 ð5Þ 6 7 6 1 76 7 6 7 6 0076 7 6 e23 7 6 2G 76 s23 7 6 7 6 23 76 7 6 7 6 1 76 7 6 7 6 76 7 6 7 6 Sym 0 76 7 6 e31 7 6 2G31 76 s31 7 6 7 4 56 7 4 5 1 4 5 2G12 e12 s12 Nonlinear elasticity in large deformations is described in Section 2.2, with applications for porous materials in Section 2.3 and for elastomers in Section 2.4. Thermoelasticity takes into account the stresses and strains induced by thermal expansion with dilatation coefficient a. For small variations of temperature y for which the elasticity parameters may be considered as constant: 1 þ n n e ¼ s À s d þ ayd ð6Þ ij E ij E kk ij ij For large variations of temperature, E; n; and a will vary. In rate formulations, such as are needed in elastoviscoplasticity, for example, the 2.1 Introduction to Elasticity and Viscoelasticity 73 derivative of E; n; and a must be considered. 1 þ n n ’ @ 1 þ n @ n @a ’ e’ ¼ s’ À s’ d þ ayd þ s À s d þ yd y ij E ij E kk ij ij @y E ij @y E kk ij @y ij ð7Þ Viscoelasticity considers in addition a dissipative phenomenon due to ‘‘internal friction,’’ such as between molecules in polymers or between cells in wood. Here again, isotropy, linearity, and small strains allow for simple models. Quadratric functions for the state potential and the dissipative potential lead to either Kelvin-Voigt or Maxwell’s models, depending upon the partition of stress or strains in a reversible part and in an irreversible part. They are described in detail for the one-dimensional case in Section 2.5 and recalled here in three dimensions. Kelvin-Voigt model: ’ ’ sij ¼ lðekk þ ylekkÞdij þ 2mðeij þ ymeijÞð8Þ Here l and m are Lame’s coefficients at steady state, and yl and ym are two time parameters responsible for viscosity. These four coefficients may be identified from creep tests in tension and shear. : Maxwell model ’ 1 þ n s n skk eij ¼ s’ ij þ À s’ kk þ dij ð9Þ E t1 E t2 Here E and n are Young’s modulus and Poisson’s ratio at steady state, and t1 and t2 are two other time parameters. It is a fluidlike model: equilibrium at constant stress does not exist. In fact, a more general way to write linear viscoelastic constitutive models is through the functional formulation by the convolution product as any linear system. The hereditary integral is described in detail for the one-dimensional case, together with its use by the Laplace transform, in Section 2.5. Z t Xn dskl p eijðtÞ ¼ Jijklðt À tÞ dt þ Jijklðt À tÞDskl ð10Þ o dt p¼1 Âà p JðtÞ is the creep functions matrix, and Dskl are the eventual stressÂà steps. The dual formulation introduces the relaxation functions matrix R Z ðtÞ Xn t de s ¼ R ðt À tÞ kl dt þ R Dep ijðtÞ ijkl t ijkl kl ð11Þ o d p¼1 When isotropy is considered the matrix, ½J and ½R each reduce to two functions: either JðtÞ, the creep function in tension, is identified from a creep 74 Lemaitre test at constant stress; JðtÞ ¼ eðtÞ=s and K, the second function, from the creep function in shear. This leads to Ds Ds e ¼ðJ þ KÞ ij À K kk d ð12Þ ij Dt Dt ij where stands for the convolution product and D for the distribution derivative, taking into account the stress steps. Or MðtÞ, the relaxation function in shear, and LðtÞ, a function deduced from M and from a relaxation test in tension RðtÞ ¼ sðtÞ=e; LðtÞ ¼ MðR À 2MÞ=ð3M À RÞ Dðe Þ De s ¼ L kk d þ 2M ij ð13Þ ij Dt ij dt All of this is for linear behavior. A nonlinear model is described in Section 2.6, and interaction with damage is described in Section 2.7. CHAPTER 2.2 Background on Nonlinear Elasticity R. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK Contents 2.2.1 Validity . ................................75 2.2.2 Deformation. ................................75 2.2.3 Stress and Equilibrium . ..................77 2.2.4 Elasticity . ................................77 2.2.5 Material Symmetry . .........................78 2.2.6 Constrained Materials . ..................80 2.2.7 Boundary-Value Problems . ..................82 References. .......................................83 2.2.1 VALIDITY The theory is applicable to materials, such as rubberlike solids and certain soft biological tissues, which are capable of undergoing large elastic deformations. More details of the theory and its applications can be found in Beatty [1] and Ogden [3]. 2.2.2 DEFORMATION For a continuous body, a reference configuration, denoted by Br, is identified and @Br denotes the boundary of Br. Points in Br are labeled by their position vectors X relative to some origin. The body is deformed quasi- statically from Br so that it occupies a new configuration, denoted B, with Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 75 76 Ogden boundary @B. This is the current or deformed configuration of the body. The deformation is represented by the mapping w : Br ! B, so that x ¼ vðXÞ X 2 Br; ð1Þ where x is the position vector of the point X in B. The mapping v is called the deformation from Br to B, and v is required to be one-to-one and to satisfy appropriate regularity conditions. For simplicity, we consider only Cartesian coordinate systems and let X and x, respectively, have coordinates Xa and xi, where a; i 2f1; 2; 3g, so that xi ¼ wiðXaÞ. Greek and Roman indices refer, respectively, to Br and B, and the usual summation convention for repeated indices is used. The deformation gradient tensor, denoted F, is given by F ¼ Grad x Fia ¼ @xi=@Xa ð2Þ Grad being the gradient operator in Br. Local invertibility of v and its inverse requires that 05 J det F51ð3Þ wherein the notation J is defined. The deformation gradient has the (unique) polar decompositions F ¼ RU ¼ VR ð4Þ where R is a proper orthogonal tensor and U, V are positive definite and symmetric tensors. Respectively, U and V are called the right and left stretch tensors. They may be put in the spectral forms X3 X3 ðiÞ ðiÞ ðiÞ ðiÞ U ¼ liu u V ¼ liv v ð5Þ i¼1 i¼1 ðiÞ ðiÞ ðiÞ where v ¼ Ru ; i 2f1; 2; 3g, li are the principal stretches, u the unit eigenvectors of U (the Lagrangian principal axes), vðiÞ those of V (the Eulerian principal axes), and denotes the tensor product.
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