Analysis of Deformation

Chapter 7: Constitutive Equations Definition: In the previous chapters we’ve learned about the definition and meaning of the concepts of stress and strain. One is an objective measure of load and the other is an objective measure of deformation. In fluids, one talks about the rate-of-deformation as opposed to simply strain (i.e. deformation alone or by itself). We all know though that deformation is caused by loads (i.e. there must be a relationship between stress and strain). A relationship between stress and strain (or rate-of-deformation tensor) is simply called a “constitutive equation”. Below we will describe how such equations are formulated. Constitutive equations between stress and strain are normally written based on phenomenological (i.e. experimental) observations and some assumption(s) on the physical behavior or response of a material to loading. Such equations can and should always be tested against experimental observations. Although there is almost an infinite amount of different materials, leading one to conclude that there is an equivalently infinite amount of constitutive equations or relations that describe such materials behavior, it turns out that there are really three major equations that cover the behavior of a wide range of materials of applied interest. One equation describes stress and small strain in solids and called “Hooke’s law”. The other two equations describe the behavior of fluidic materials. Hookean Elastic Solid: We will start explaining these equations by considering Hooke’s law first. Hooke’s law simply states that the stress tensor is assumed to be linearly related to the strain tensor. More specifically that every strain component depends on all independent stress components or vice versa (i.e. every stress component depends on all strain components). Such a relationship can be written concisely as: σ = ij Cijkl ekl σ where ij is the stress tensor, ekl is the strain tensor, and Cijkl is a fourth-order tensor whose components are essentially proportionality coefficients (which is clearer upon expansion of coefficients). Such constants are also called the “elastic constants” or “moduli” of the solid material and are independent of the magnitude of stress or strain at a material point. The question might arise: How does one arrive at such a relationship? The answer is by looking at some known phenomena and extrapolating on it to cover more general behavior of material deformation. Specifically and more simply, consider a rod/bar of material cut out from a board of the same material along one direction (call it the x- direction). Alternatively, one may have cut out this rod along the y-direction and so forth. When one applies a tensile or extension force, say at the ends of this rod, we all know from elementary experience and scientific knowledge that the stress σ (force/end area) is 1 proportional to the strain e (change in length/length) via a proportionality constant, call it E, such that we can write σ = Ee Indeed, a lot of us know this equation as Hooke’s law and call the constant E either Young’s modulus, modulus of elasticity, or the stiffness of the bar. The truth is that this last equation is a special simple 1-D form of the true Hooke’s law given by the tensorial and more general relationship above (which is inherently 3-D). At this juncture, it is worth noting that Hooke’s law in this form is the result of experimental observation especially valid for small strains in the material. To be more precise, and utilizing knowledge from earlier chapters, σ here is really σxx and e is really exx. Because of the specific directionality of the problem, and hence the x subscripts, E can also be written more appropriately as Ex (i.e. we should have written σ = xx Exexx ) The reason for Ex having a special value when performing this tension test along the x-axis, is that it can be envisioned in the most general case that if we pull on the rod in any other directions, say along y-direction, then we must have a different proportionality constant, i.e. E ≠ Ex. Indeed, thinking generally we can write in this σ = particular case yy E y eyy . The same argument can be made for the z-direction, for example. Ex, Ey and Ez are elastic constants in the sense above, i.e. they relate stresses to strain in the material. These three elastic constants are indeed related to the constants of the fourth-order tensor Cijkl as well be illustrated later (by the way Cijkl is also called the “stiffness tensor”). F F x Figure: a bar in tension In the above light, one can see that the general and true form of Hooke’s law is simply an envisioned generalization and extension of the 1-D form where it is assumed that stresses (whether normal or shear) are related (or coupled) not just to one component of strain (normal or shear) but to all of them in general (and visa versa, every strain component is related to all six independent stress components in general). The exact form of the constitutive equation (Hooke’s law for example with linear dependence on strain and not nonlinear quadratic or other dependence) depends on the material and the assumption used in constructing the equation. Upon expansion of indices of the tensorial equation above, one gets σ = + + + + + + + + ij Cij11e11 Cij22e22 Cij33e33 Cij12e12 Cij13e13 Cij23e23 Cij21e21 Cij31e31 Cij32e32 For example, 2 σ = + + + + + + + + 11 C1111e11 C1122e22 C1133e33 C1112e12 C1113e13 C1123e23 C1121e21 C1131e31 C1132e32 Notice that barring any special conditions or restriction, Cijkl has 3×3×3×3 = 81 components or elastic constants to be determined for a general material. This is a rather large number of constants that no real material has and can be reduced, based on physical arguments, as follows. First, eij is a symmetric tensor, i.e. eij = eji. Hence, we can write σ = = = ij Cijklekl Cijlk elk Cijlk ekl − = ⇒ ekl (Cijkl Cijlk ) 0 = ⇒ Cijkl Cijlk Based on the last relationship, Cijkl now has 3×3×6 = 54 independent constants. σ σ = σ Similarly, ij is symmetric, i.e. ij ji , hence σ = = σ = ij Cijklekl ji C jiklekl − = ⇒ ekl (Cijkl C jikl ) 0 = ⇒ Cijkl C jikl This reduces the number of independent constants to 6×6 = 36. Note that since Cijkl is a fourth-order tensor, it transforms accordingly: ′ = β β β β Cijkl im jn ks lt Cmnst Since Cijkl has only 36 independent constants, we can conveniently then write Hooke’s law in matrix notation as: σ C C C C C C e 11 1111 1122 1133 1123 1113 1112 11 σ C C C C C C e 22 2211 2222 2233 2223 2213 2212 22 σ 33 C3311 C3322 C3333 C3323 C3313 C3312 e33 = σ 23 C2311 C2322 C2333 C2323 C2313 C2312 2e23 σ C C C C C C 2e 13 1311 1322 1333 1323 1313 1312 13 σ 12 C1211 C1222 C1233 C1223 C1213 C1212 2e12 where e11 e11 e e 22 22 e33 e33 = γ 2e23 23 γ 2e13 13 γ 2e12 12 , and where the 6×6 matrix of elastic constants is also called the “stiffness matrix” C. 3 Further reductions in the number of independent elastic constants from 36 can be achieved as follows: The work dU done by the stress components σij acting on a unit cube of elastic material when the deformation is increased so that the strain tensor components increase by deij, this work is given by: = σ dU ij deij (which has units of work per unit volume) The total work in the process would be given by U: eij U = σ de ∫ ij ij = eij 0 In 1-D, this is the area under the stress-strain line or curve: 1-D stress-strain line σ11 Area = U e11 de11 The quantity U is also called the “strain energy density” since its units are energy units per unit volume. From the last equation, one can see that ∂U = σ ∂ ij eij = σ σ = Now since dU ij deij and ij Cijkl ekl ∂U ⇒ dU = C e de = de ijkl kl ij ∂ ij eij ∂U ⇒ = C e ∂ ijkl kl eij Differentiating the last equation gives: ∂ ∂U = C ∂ ∂ ijkl ekl eij Interchanging indices order in the last equation gives: ∂ ∂U = C ∂ ∂ klij eij ekl From the last two equations, one can see that: = Cijkl Cklij 4 The last equation means that the stiffness matrix C is symmetric. This further reduces the number of independent elastic constants to 21 from 36. A material with 21 independent constants in Cijkl is called an “anisotropic material” or a “generally-anisotropic material”. A “Monoclinic” material is a material that exhibits symmetry with respect to one plane. We take this plane to be the x1x2 plane without loss of generality. The elastic constants of the material do not change under the coordinate change: ′ = ′ = ′ = − x1 x1 , x2 x2 , x3 x3 x 3 ′ x2 , x2 ′ x1 , x1 ′ x3 The transformation matrix in this case is: 1 0 0 β = (β ) = 0 1 0 ij 0 0 −1 Under this transformation, we must have: ′ = Cijkl Cijkl Check: C′ = β β β β C 1111 1i 1 j 1k 1l ijkl ′ = β β β β = ⇒ C1111 11 11 11 11C1111 C1111 The above relationship is satisfied for any arbitrary C1111 . ′ Consider now C1123 . It is supposed to be equal to C1123 under the transformation above, ′ = i.e. C1123 C1123 Check: C′ = β β β β C 1123 1i 1 j 2k 3l ijkl ′ = β β β β = − ⇒ C1123 11 11 22 33C1123 C1123 ′ Since C1123 can NOT be BOTH equal to and the negative of C1123 UNLESS = = ′ = C1123 0 ⇒ C1123 C1123 0 By similar arguments, we can show that 7 additional constants are equal to zero so that the number of elastic constants drops to 13 from the 21 describing a generally-anisotropic material: 5 C C C 0 0 C 1111 1122 1133 1112 C2211 C2222 C2233 0 0 C2212 C3311 C3322 C3333 0 0 C3312 C = 0 0 0 C2323 C2313 0 0 0 0 C C 0 1323 1313 C1211 C1222 C1233 0 0 C1212 An “orthotropic” material has symmetry of its elastic properties with respect to two orthogonal planes.

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