ES 240Solid MechanicsZ. Suo

**Finite Deformation**

References

匡震邦, 非线性连续介质力学, 上海交通大学出版社, 2002. My favorite textbook on nonlinear continuum mechanics, written by the man who taught me the subject in 1985.

G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, 2000.

黄筑平，连续介质力学基础，高等教育出版社， 2003.

T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, 2000.

L.E. Malvern, Introduction to the Mechanics of Continuous Medium, Prentice-Hall, 1969.

C. Trusdell and W. Noll, The Non-linear Field Theoriesof Mechanics, 3rd edition, Springer, 2004.

C. Trusdell and R.A. Toupin, The Classical Field Theories, in Encyclopedia of Physics, Volume III/1, Pringer-Verlag, Berlin, 226-793 (1960).

L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd ed. 1975. Reissued in 2005.

**Be wise, linearize.** Following the advice of George Carrier, we have beenmostly looking at Hookean materials and infinitesimal deformation. We have mixed the 3 ingredients in solid mechanics (deformation geometry, momentum balance, material law)without fussing over subtleties. The results arefascinating and useful. Now we wish to go nonlinear, hopefully also with wisdom. We will refine the ingredients by considering non-Hookean materials and finite deformation. The two refinements need not be mixed. For example, a viscoelastic material is non-Hookean, but deformation of such a material can be infinitesimal. In this brief introduction to finite deformation, we will outline some of the fundamental considerations, and describe a few illustrative phenomena.

**Finite deformation.** When a structure deforms, Newton’s law holds true in every deformed state. We have often violated this law. For example, in analyzing a truss, we have balanced forces as if the truss did not deform.

You might think that a structure sufferinga small strain, say less than 1%, entitles you to neglect the change in geometry when you balance forces. A counter example is familiar to you. Upon buckling, the strains in a column are indeed small, but you must enforce equilibrium in the deflected state of the column. Mechanics of deformation is a tricky business. We proceed with caution. The essential point is this: we must enforce Newton’s law in every deformed state, and justify any simplification on this basis.

**Non-Hookean materials. **Moving nonlinear and inelastic will take usin many directions. For example, when deformation is large, the force may vary nonlinearlywith the elongation. As another example, we’ve already looked at time-dependent behavior of materials, such as viscoelasticity. wealso have daily experience of metals. After elastic deformation, upon unloading, a metal recovers its shape. After plastic deformation, upon unloading, the metal does not fully recover its shape. If we apply an axial force to a metal bar, and measure its length, the force-length relation is linear for elastic deformation, and is nonlinear for plastic deformation. During unloading, the metal bar deforms elastically. After plastic loading and elastic unloading, the force-length relation is not a one-to-one relation, but ishistory-dependent. Of course, a viscoelastic material is also history-dependent.

*To analyze finite, history-dependent deformation of a structure, a general approach is to evolvethe stateincrementally, and enforce Newton’s law in every state.*

**A rod under axial load. **We also proceed with our subject incrementally, beginning with a rod in incremental states of uniaxial stress. Initially, the rod is unstressed, and has cross-sectional area and length . The rod is then subject to an axial force P, and deforms to cross-sectional area a and length l.We next examine the 3 ingredients in solid mechanics.

**Strain measures.** Any state can be used as a reference state. For example, we can take the initial, unstressed state as the reference state. Define the **engineering strain** by the elongation divided by the reference length:

.

Another strain measure is defined as follows. Deform the material from a current length l by a small amount to . Define the increment in the strain, , as the increment in the length of the rod divided by the current length of the rod, namely,

.

This equation defines the increment of natural strain. Integrating from to l, we obtain that

.

Yet one more strain measure, the Lagrange strain, is defined as

.

This definition is hard to motivate in 1D. But if you take the view that any increasing function of is a suitable measure of strain, then no motivation is really needed.

Indeed, even the ratio itself has a name: the stretch is defined as the length of the rod in the current state divided by the length of the rod in the reference state:

.

There seems to be no lack of human ingenuity to form a dimensionless quantity out of two lengths L and l. Needless to say, all these strain measures contain the same information. For example, every one of the measures defined above is an increasing function of stretch:

.

Because they are all one-to-one functions, any one measure can be taken to be “basic” and then used to express all the rest. For example, we can express all measures in terms of the engineering strain:

Thus, when we call e the engineering strain, we do not mean that e is unscientific or crude or unnatural strain. We just need a name. When the strain is small, namely, , the three measures are approximately equal,.

Later on, we will provide motivations for some of these definitions, but these motivations are probably elaborate ways to express preferences of individual people.

**Stress measures. Work done by a force.** When dealing with finite deformation, we must be specific about the area used in defining the stress. Define the nominal stress, s, as the force in the current state divided by the area in the reference state:

.

When the rod elongates from length l to length , the force P does work . Recall that and , so that the work done by the force is

.

Since AL is the volume of the rod in the reference state, we note that

.

We say that the nominal stress and the engineering strain are work-conjugate. Also note that . Consequently, the stretch is also work-conjugate to the nominal stress.

Define the true stress, , as the force in the current state divided by the area in the current state, namely,

.

Recall that . The work done by the force is

.

Since is the current volume of the bar, we note that

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That is, the true stress is work-conjugate to the natural strain.

Given a measure of strain, we can define its work-conjugate stress. For example, consider the Lagrange strain, . Subject to an increment in the strain, , the force acting on the element does the work. Denote

.

This expression defines a new measure of stress, S.

This new stress measure does not have a “simpler” interpretation than its status as the work conjugate to the Lagrange strain. Indeed, if we are liberal about the definition of strain measures, without being obsessive about “motivating” each measure, we may as well take a liberal view to call the work conjugate of each strain a stress measure, and name the stress measure after a mechanician who can no longer protest. You can easily invent and name other stress measures, but the above stress measures have already got names:

- : true stress or the Cauchy stress.
- : nominal stress or the first Piola-Kirchhoff stress.
- : the second Piola-Kirchhoff stress.

Recall the relations among the measures of strain:

We obtain the relations among their increments:

.

Consequently, the three measures of stress are related as

.

All stress measures are linear in force applied to the cross-sectional area, but differ by geometric factors.

Material laws. For a metal undergoing large, plastic deformation, the stress-strain curve (without unloading) is often fit to a power lawin terms of the true stress and the natural strain:

where K and are parameters to fit experimental data. Some representative values: N = 0.15-0.25 for aluminum, N = 0.3-0.35 for copper, N = 0.45-0.55 for stainless steel. Khas the dimension of stress; it represents the true stress at strain . Representative values for K are 100 MPa – 1GPa. At large deformation, volumetric strain is negligible compared to tensile strain. Consequently, the material is often taken to be incompressible.

Rubbers are often assumed to obey the neo-Hookean law (more details later). For a rubber rod in uniaxial states of stress, the stress-strain data are fit to

.

Recall that . For small strains, , the above reduces to

.

Thus, we interpret as Young’s modulus and the shear modulus. Rubbers are nearly incompressible, so that Poisson’s ratio is taken to be ½. Representative values for are 1 MPa – 100 MPa.

**Are these alternatives necessary?**Now we have described the 3 ingredients for a rod under uniaxial tension. Even in this simplest setting, for each ingredient we have given several alternative descriptions. Some alternatives are necessary; for example, metals and rubbers behave differently. But the difference in their force-displacement relations does not justify us to use different stress and strain measures to describe different materials. In fact, to see the difference in material behavior, we would like to use the same stress and strain measures for both materials. For example, we can use the natural strain to describe the stress-strain relation for rubbers:

.

This change of variable immediately brings out a key insight: in tension, the stress in rubbers rises more steeply than in metals. We will return to this insight shortly.

Are these alternative stress and strain measures necessary? I have my own thoughts, but you should form your own opinions. The question perhaps boil down to something no more profound than asking, “Is it necessary to know many alternative roads to Boston Common?” Whatever your opinions are, however, it may alleviate some of your pains in studying the subject by knowing that textbooks of nonlinear continuum mechanics are full of equivalent alternatives at every turn. These alternatives often hide behind forests of notation and verbiage, and may offer some tantalizing sights. You will just have to look beyond them for matters of consequence.

Exercise. Use the 3 ingredients outlined about to obtain the force-deflection relation for the truss sketched in the beginning of the notes. Assuming all three members of the truss are made of rubber bands, and that deformation is large.

**Necking in a bar. Considère condition.** Let us try to apply the newly refined 3 ingredients to a specific phenomenon: necking. Subject to a tensile force, a metal bar first elongates uniformly and then, at some strain, a small part of the rod starts to thin down preferentially, forming a neck. By contrast, a rubber band under tension usually does not form a neck. We would like to interpret these observations. To do so we must be explicitly specify the measures of stress and strain.

Here is a summary of the 3 ingredients, using a specific set of alternatives:

- Force balance:
- Material law: For a metal bar under uniaxial tension, the true stress relates to the natural strain as .
- Deformation geometry:. We will assume that the volume of the rod is constant during deformation, , or.

Put the three ingredients together, and we obtain the force as a function of strain:

.

Plot P as a function of . In plotting the figure, I’ve set , with N = 0.5. Observe the two competing factors: material hardening and geometric softening. As the bar elongates, the material hardens, as reflected by the hardening exponent in the stress-strain relation . At the same time, the elongation reduces the cross-sectional area, an effect known as geometric softening. For small deformation, as ; material hardening prevails, and the force increases as the bar elongates. For large deformation, so long as the stress-strain relation increases slower than , as , geometric softening prevails, and the force drops as the bar elongates.

To determine the peak force, note that

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Consequently, the force P peaks when the true stress equals the tangent modulus:

.

This equation, known as the Considère condition, determines the strain at which the force peaks. For the power-law material, the force peaks at the critical strain

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When the metal bar is loaded beyond this critical strain, deformation becomes nonuniform, with a segment of the bar elongates at a higher strain than the rest of the bar. That is, a neck forms in the bar. For an analysis of the nonuniform deformation, see Needleman (1972, A numerical study of necking in circular cylindrical bar, Journal of the Mechanics and Physics of Solids, 20, 111). ABAQUS can be used to study the necking process.

For a rubber band, assume the material is Neo-Hookean:

.

Thus, at a large tensile strain, the true stress increases exponentially with the natural strain, so that the force always increases with the strain. The rubber band will not form a neck under uniaxial tension.

**Hyperelastic materials**. We next explore nonlinear stress-strain relations under multiaxial states of stress. Of course, the only way to really know such relations is to run tests for a given history of state of stress, but this would be too time-consuming and quickly become impractical. We’ll have to reduce the number of tests by some approximations. The art of making such compromise between accuracy and labor is known as formulating constitutive laws. As an example, here we attempt to describe this art for rubbers.

Arubberrod, length L and cross-sectional area A in the unstressed state, is stretched by force Pto length and cross-sectional area a. When the rod extends from length l to length , the force does work . Recall that and , so that the work is

.

SinceAL is the volume of the rod in the undeformedstate, we note that

.

We say that the nominal stress is work-conjugate to the stretch. Define the nominal energy density by

.

Assuming that the work done by the force is fully stored as energy in the rod, we obtain that

.

Note that Wshould be considered the Helmholtz free energy. We’ll only consider the isothermal conditions, so that that we’ll drop the dependence on temperature.

We can measure experimentally the nominal stress as a function of stretch, , and then integrate the curve to obtain , , known as the strain-energy function. Alternatively, we can obtain an expression of from some theoretical considerations, and then obtain by

.

For anisotropic material, we can represent a material particle by a rectangular block cut in the orientation of the three principal stresses. Let the blockbe stretched in the three directions by1, 2 and 3, and the corresponding nominal stresses bes1, s2 and s3. The complete stress-strain relations involve three functions , and . In general, we can run tests to determine these functions. However, as we indicated above, running test alone would be too time-consuming. Instead, we will formulate a constitutive law on the basis of the assumption that work done by the forces is all stored in the block as energy.

When the stretches increase by, and , following the same line of reasoning as given above, we conclude that the free energy changes by

.

Now the strain energy is a function of the three stretches,. Once the function is determined by experiment, say, the stresses are calculated from the partial derivatives:

.

An elastic material whose stress-strain relation is derivable from a strain-energy function is known as a hyperelastic material. An elastic material whose stress-strain relation is not derivable from a strain-energy function is called a hypoelastic material.

A rectangular block of a material, lengths , and in the undeformed state, is subject to forces , and on its faces, and is deformed into a block of lengths , and . The nominal stress on one face is

.

The true stress on the same face is

.

Consequently, the true stress relates to the nominal stress by

.

The true stress is derivable from the strain-energy function:

.

The other two true stress components can be similarly obtained.

**Incompressible, isotropic, hyperelastic material.** When a material undergoes large deformation, the amount of volumetric deformation is often small compared to the overall deformation. Consequently, we may neglect the volumetric deformation, and assume that the material is incompressible.

A block of a material, of lengths , and in the undeformed state, is deformed into a rectangle of lengths , and . If the material is incompressible, the volume of the block must remain unchanged, namely,, or

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The incompressibility places a constraint among the three stretches: they cannot vary independently. Wemay regard and as independent variables, so that

.

Consequently, the strain-energy density is a function of two independent variables, .Consequently, only biaxial tests need be done to fully characterize the material.

Inserting the constraint into the expression , we obtain that

.

Once the function is determined, the stress-stretch relation is given by differentiation:

, .

These relations, together with the incompressibility condition , replace Hooke’s law and serve as the stress-strain relations for incompressible, isotropic and hyperelastic materials.

For incompressible materials, the true stresses relate to the nominal stresses as

.

Consequently, the stress-strain relations become

, .

**Stress-strain relations for rubbers.** The function can be determined by subjecting a sheet of rubber under biaxial stress states. The form of the function is sometimes inspired by theoretical considerations. Here are three often used forms.