(so that any vector a associated with the current configuration is written as $\mathbf { a } = a ^ { \alpha } \mathbf { e } _ { \alpha } )$ and ${ \bf E } _ { \alpha }$ , $\alpha = 1 , 2 , 3$ , as the basis at the same material point but in the reference configuration. (Since we assume that both of these basis systems are adequate to express any vector-valued function by its components in the basis system--that is, the basis vectors are not linearly dependent--either would, in principal, serve for both configurations. We introduce two distinct systems by preference, because each is chosen as particularly suitable for a particular configuration.) Since we do not yet impose any particular restrictions on the $\mathbf { e } _ { \alpha }$ or the ${ \bf E } _ { \alpha }$ (except for the requirement that the vectors must not be linearly dependent), we cannot assume that they will be orthogonal or of unit length: we will, therefore, need to use the corresponding contravariant vectors defined by $$ \mathbf {e} ^ {\alpha} \cdot \mathbf {e} _ {\beta} = \delta_ {\beta} ^ {\alpha} \quad \mathrm{and} \quad \mathbf {E} ^ {\alpha} \cdot \mathbf {E} _ {\beta} = \delta_ {\beta} ^ {\alpha}, $$ and the contravariant metric tensors $$ g ^ {\alpha \beta} = \mathbf {e} ^ {\alpha} \cdot \mathbf {e} ^ {\beta} \quad \mathrm{and} \quad G ^ {\alpha \beta} = \mathbf {E} ^ {\alpha} \cdot \mathbf {E} ^ {\beta}. $$ We can express the deformation gradient, F, numerically by projecting it onto the bases: Equation 1.4.1-13 $$ \mathbf {F} = \mathbf {e} ^ {\alpha} F _ {\alpha \beta} \mathbf {E} ^ {\beta}. $$ Recall the definition of F: $$ d \mathbf {x} = \mathbf {F} \cdot d \mathbf {X} \left(= \frac {\partial \mathbf {x}}{\partial \mathbf {X}} \cdot d \mathbf {X}\right). $$ Since the components of dx along $\mathbf { e } _ { \alpha }$ are $d x _ { \alpha } = d \mathbf { x } \cdot \mathbf { e } _ { \alpha }$ and we can write $d \mathbf { X } = d X ^ { \beta } \mathbf { E } _ { \beta }$ , $$ \begin{array}{l} d x _ {\alpha} = \mathbf {e} _ {\alpha} \cdot \mathbf {F} \cdot \mathbf {E} _ {\beta} d X ^ {\beta} \\ = \mathbf {e} _ {\alpha} \cdot \frac {\partial \mathbf {x}}{\partial X ^ {\beta}} d X ^ {\beta}. \\ \end{array} $$ Thus, writing $d x _ { \alpha } = F _ { \alpha \beta } d X ^ { \beta }$ defines $$ F _ {\alpha \beta} = \mathbf {e} _ {\alpha} \cdot \mathbf {F} \cdot \mathbf {E} _ {\beta} = \mathbf {e} _ {\alpha} \cdot \frac {\partial \mathbf {x}}{\partial X ^ {\beta}}. $$ We must continue to bear in mind that the first index of $F _ { \alpha \beta }$ is associated with a component of F along a base vector in the current configuration $\left( \mathbf { e } _ { \alpha } \right.$ in this case), while its second index is associated with a component of F along a base vector in the reference configuration $\left( \mathbf { E } _ { \beta } \right)$ . From Equation 1.4.1-13 we can write $$ \begin{array}{l} \mathbf {F} \cdot \mathbf {F} ^ {T} = \mathbf {e} ^ {\alpha} F _ {\alpha \beta} \mathbf {E} ^ {\beta} \cdot \mathbf {E} ^ {\gamma} F _ {\delta \gamma} \mathbf {e} ^ {\delta} \\ = \mathbf {e} ^ {\alpha} F _ {\alpha \beta} G ^ {\beta \gamma} F _ {\delta \gamma} \mathbf {e} ^ {\delta}, \\ \end{array} $$ where $G ^ { \alpha \beta }$ is the contravariant metric of the basis system that we have chosen in the reference configuration. The eigenproblem for the squared principal stretch ratios and their directions is solved by finding the eigenvalues of the matrix of numbers $F _ { \alpha \beta } G ^ { \beta \gamma } F _ { \delta \gamma }$ . The eigenvectors will appear as the components $( n _ { I } ) _ { \alpha }$ along the $\mathbf { e } ^ { \alpha }$ base vectors in the current configuration. Since we have defined the left stretch on the current configuration as $$ \mathbf {V} = \lambda_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \lambda_ {I I} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}, $$ we will write its components on the basis in the current configuration as $$ V _ {\alpha \beta} = \lambda_ {I} n _ {I \alpha} n _ {I \beta} + \lambda_ {I I} n _ {I I \alpha} n _ {I I \beta} + \lambda_ {I I I} n _ {I I I \alpha} n _ {I I I \beta} = \mathbf {e} _ {\alpha} \cdot \mathbf {V} \cdot \mathbf {e} _ {\beta}; $$ and, since $$ \mathbf {V} ^ {- 1} = \frac {1}{\lambda_ {I}} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \frac {1}{\lambda_ {I I}} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \frac {1}{\lambda_ {I I I}} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}, $$ $$ (V ^ {- 1}) _ {\alpha \beta} = \frac {1}{\lambda_ {I}} n _ {I \alpha} n _ {I \beta} + \frac {1}{\lambda_ {I I}} n _ {I I \alpha} n _ {I I \beta} + \frac {1}{\lambda_ {I I I}} n _ {I I I \alpha} n _ {I I I \beta}. $$ The polar decomposition gives $$ \begin{array}{l} \mathbf {R} = \mathbf {V} ^ {- 1} \cdot \mathbf {F} \\ = \mathbf {e} ^ {\alpha} (V ^ {- 1}) _ {\alpha \gamma} \mathbf {e} ^ {\gamma} \cdot \mathbf {e} ^ {\delta} F _ {\delta \beta} \mathbf {E} ^ {\beta} \\ = \mathbf {e} ^ {\alpha} (V ^ {- 1}) _ {\alpha \gamma} g ^ {\gamma \delta} F _ {\delta_ {\beta}} \mathbf {E} ^ {\beta}, \\ \end{array} $$ so $$ R _ {\alpha \beta} = (V ^ {- 1}) _ {\alpha \gamma} g ^ {\gamma \delta} F _ {\delta_ {\beta}}, $$ where $g ^ { \gamma \delta }$ is the contravariant metric tensor of the basis system we have chosen to use in the current configuration and--as with $F _ { \alpha \beta } \mathrm { - - } \mathbf { W } \mathbf { e }$ see that the first index of $R _ { \alpha \beta }$ is associated with the contravariant base vector $\mathbf { e } ^ { \alpha }$ in the current configuration, while the second index is associated with the contravariant base vector $\mathbf { E } ^ { \beta }$ in the reference configuration. We should take care to understand the distinction between the direct matrix expression of the rigid body rotation of the principal directions of strain of the material, R, and the components of R expressed on a particular basis. Suppose, for example, that the rigid body rotation at a point is zero (that is, R = I) but we, nevertheless, have chosen different basis systems $\mathbf { e } _ { \alpha }$ and ${ \bf E } _ { \alpha }$ . In this case $R _ { \alpha \beta } = \mathbf { e } _ { \alpha } \cdot \mathbf { R } \cdot \mathbf { E } _ { \beta } = \mathbf { e } _ { \alpha } \cdot \mathbf { I } \cdot \mathbf { E } _ { \beta } = \mathbf { e } _ { \alpha } \cdot \mathbf { E } _ { \beta }$ . This implies that, even though R is a unit matrix (in the sense that operating on any vector with this matrix makes no change in that vector), the numerical values we have chosen to store the matrix--the $R _ { \alpha \beta } { \bf - - } { \bf d } _ { 0 }$ not form a unit matrix of numbers unless the $\mathbf { e } _ { \alpha }$ and the ${ \bf E } _ { \alpha }$ are coincident and orthonormal. Thus, our choice of quite general basis systems that are not the same in the current and reference configurations (introduced as being "natural" for writing results for shells) somewhat complicates the interpretation of the numbers we store. In the previous few paragraphs we have chosen to explore the expression of the basic results we have derived so far for the kinematics of the total motion in terms of quite general basis systems, ${ \bf e } _ { \alpha }$ and $\mathbf { E } _ { \alpha }$ . In ABAQUS we wish to express results as simply and directly as possible, and we can do so by choosing particular sets of basis vectors that offer the most convenience for our purposes. First, we take the ${ \bf e } _ { \alpha }$ (and, by extension, the ${ \bf E } _ { \alpha }$ , since these are just the ${ \bf e } _ { \alpha }$ at the beginning of the motion) to be a local, orthonormal system at each point. Although it is not possible to construct a Cartesian system with orthonormal base vectors over a general shell surface, we can always project the general results onto such a system when that system is chosen specifically at each point where we need to make the projection--typically at the integration points of the elements. The choice of which system is used as this local orthonormal basis is made in ABAQUS at two levels: we distinguish continuum ("solid") elements from structural (shell and beam) elements, and we distinguish the default choice of directions from the particular choice of directions indicated by the user when the \*ORIENTATION option is included. For continuum elements the default $\mathbf { E } _ { \alpha }$ are unit vectors along the axes of the global Cartesian system chosen for the problem. At points where the \*ORIENTATION option is invoked, the ${ \bf E } _ { \alpha }$ are those specified in that option. For shells (and membranes) we take ${ \bf E } _ { 1 }$ and $\mathbf { E } _ { 2 }$ tangent to the shell's reference surface and $\mathbf { E } _ { 3 }$ normal to that surface at the point under consideration. Without \*ORIENTATION $\mathbf { E } _ { 1 }$ is the projection of the global x-axis onto the reference surface $\mathbf { o r } ,$ if the global x-axis is almost normal to that surface at the point, $\mathbf { E } _ { 1 }$ is the projection of the global z-axis onto the surface. With \*ORIENTATION $\mathbf { E } _ { 1 }$ and $\mathbf { E } _ { 2 }$ are the projections of two axes specified in the \*ORIENTATION option onto the reference surface at the point. In all cases $\mathbf { E } _ { 3 }$ is normal to the shell's reference surface. For beams ${ \bf E } _ { 1 }$ is along the beam axis, with $\mathbf { E } _ { 2 }$ and $\mathbf { E } _ { 3 }$ defined from the beam section definition option and beam normals given as part of the nodal coordinate definition. For continuum elements without \*ORIENTATION the same schemes are applied to define the basis system in the current configuration. For continuum elements with the \*ORIENTATION option invoked at the point and in all cases for shells, beams, and membranes, the $\mathbf { e } _ { \alpha }$ are defined by $$ \mathbf {e} _ {\alpha} = \mathbf {R} \cdot \mathbf {E} _ {\alpha}. $$ These schemes all have the same property: at any point in time the ${ \bf e } _ { \alpha }$ are orthonormal vectors: ${ \bf e } _ { \alpha } \cdot { \bf e } _ { \beta } = \delta _ { \alpha \beta }$ , so $ { \mathbf { e } } ^ { \alpha } = { \mathbf { e } } _ { \alpha }$ and, thus, $g _ { \alpha \beta } = g ^ { \alpha \beta } = g _ { \alpha } ^ { \beta } = \delta _ { \alpha \beta }$ , and--in particular-- $\mathbf { E } _ { \alpha } \cdot \mathbf { E } _ { \beta } = \delta _ { \alpha \beta }$ and, thus, $G ^ { \alpha \beta } = \delta ^ { \alpha \beta }$ . This simplifies the understanding of all the quantities we write, since the components of any tensor $T _ { \alpha \beta . }$ :: are always the physical projections of that tensor-valued quantity on the local orthogonal basis system ${ \bf e } _ { \alpha }$ and we need not distinguish covariant and contravariant components as we did in the general development above. In practical terms the only price we must pay for this simplicity is in shells when we have to use a separate basis system at each point under study, since we cannot construct a single system with the orthonormal property on a general curved surface. (In an axisymmetric system we also have to use $d x ^ { 3 } = r d \theta$ to ensure that the $\mathbf { e } _ { 3 }$ base vector is a unit vector, but this is a minor point.) The simplifications are valuable and, from our perspective of studying finite element formulations, they are bought at modest cost, since we generally only consider a single integration point at a time. Throughout the rest of this manual, whenever we need to write down particular components of a tensor, we shall assume that the basis on which they are written has the orthonormal property ${ \bf e } _ { \alpha } \cdot { \bf e } _ { \beta } = \delta _ { \alpha \beta }$ . The material also undergoes rigid body translation, but this is not important in the development since we need consider only relative motion of neighboring points because we are interested in the deformation of the material to link the kinematics of the motion to the material's constitutive behavior. Numerically, rigid body translation is significant only for two reasons. One is that the spatial discretization must allow rigid body translation without giving strain, which is important in choosing interpolation functions for the finite elements. The other is that care must be exercised to ensure that the strain and rotation are calculated accurately when the rigid body motion is large, since then the strain and rotation depend on the difference between two very large motions. # 1.4.2 Strain measures Strain measures used in general motions are most simply understood by first considering the concept of strain in one dimension and then generalizing this to arbitrary motions by using the polar decomposition theorem just derived. # Strain in one dimension We already have a measure of deformation--the stretch ratio ¸. In fact, ¸ is itself an adequate measure of "strain" for a number of problems. To see where it is useful and where not, first notice that the unstrained value of ¸ is 1.0. A typical soft rubber component (such as a rubber band) can change length by a large factor when it is loaded, so the stretch ratio ¸ would often have values of 2 or more. In contrast, a typical structural steel component will be designed to respond elastically to its working loads. Such a material has an elastic modulus of about $2 0 0 \times 1 0 ^ { 3 } \mathrm { M P a } ( 3 0 \times 1 0 ^ { 6 } \mathrm { l b } / \mathrm { i n } ^ { 2 } )$ at room temperature and a yield stress of about 200 MPa $( 3 0 \times 1 0 ^ { 3 } 1 6 / \mathrm { i n } ^ { 2 } )$ , so the stretch at yield will be about 1.001 in tension, 0.999 in compression. The stretch ratio is an unsatisfactory way of measuring deformation for this case because the numbers of interest begin in the fourth significant digit. To avoid this inconvenience, the concept of strain is introduced, the basic idea being that the strain is zero at ¸ = 1, when the material is "unstrained." In one dimension, along some "gauge length" dX, we define strain as a function of the stretch ratio, ¸, of that gauge length: $$ \varepsilon = f (\lambda). $$ The objective of introducing the concept of strain is that the function f is chosen for convenience. To see what this implies, suppose " is expanded in a Taylor series about the unstrained state: Equation 1.4.2-1 $$ \varepsilon = f (1) + (\lambda - 1) \frac {d f}{d \lambda} + \frac {1}{2 !} (\lambda - 1) ^ {2} \frac {d ^ {2} f}{d \lambda^ {2}} + \dots $$ We must have $f ( 1 ) = 0 , \mathtt { s o } \varepsilon = 0 \mathrm { a t } \lambda = 1$ (this was the main reason for introducing this idea of "strain" instead of just using the stretch ratio). In addition, we choose $d f / d \lambda = 1 \mathrm { a t } \lambda = 1$ so that for small strains we have the usual definition of strain as the "change in length per unit length." This ensures that, in one dimension, all strain measures defined in this way will give the same numerical value to the order of the approximation when strains are small (because then the higher-order terms in the Taylor series are all negligible)--regardless of the magnitude of any rigid body rotation. Finally, we require that $d f / d \lambda > 0$ for all physically reasonable values of ¸ (that, is for all $\lambda > 0 )$ so that strain increases monotonically with stretch; hence, to each value of stretch there corresponds a unique value of strain. (The choice of $d f / d \lambda > 0$ is arbitrary: we could equally well choose $d f / d \lambda < 0$ , implying that the strain is positive in compression when $\lambda < 1$ . This alternative choice is often made in geomechanics textbooks because geotechnical problems usually involve compressive stress and strain. The choice is a matter of convenience. In ABAQUS we always use the convention that positive direct strains represent tension when $\lambda > 1$ . This choice is retained consistently in ABAQUS, including in the geotechnical options.) With these reasonable restrictions ( f = 0 and $d f / d \lambda = 1$ at ¸ = 1, and $d f / d \lambda > 0$ for all $\lambda > 0 )$ , many strain measures are possible, and several are commonly used. Some examples are Nominal strain (Biot's strain): f (¸) = ¸ ¡ 1: In a uniformly strained uniaxial specimen, where l is the current and L the original gauge length, this strain is measured as $( l / L ) - 1$ . This definition is the most familiar one to engineers who perform uniaxial testing of stiff specimens. Logarithmic strain: $f ( \lambda ) = \ln \lambda .$ This strain measure is commonly used in metal plasticity. One motivation for this choice in this case is that, when "true" stress (force per current area) is plotted against log strain, tension, compression and torsion test results coincide closely. Later we will see that this strain measure is mathematically appropriate for such materials because, for these materials, the elastic part of the strain can be assumed to be small. Green's strain: $f ( \lambda ) = \frac { 1 } { 2 } \left( \lambda ^ { 2 } - 1 \right) .$ This strain measure is convenient computationally for problems involving large motions but only small strains, because, as we will show later, its generalization to a strain tensor in any three-dimensional motion can be computed directly from the deformation gradient without requiring solution for the principal stretch ratios and their directions. All of these strains satisfy the basic restrictions. Obviously many strain functions are possible: the choice is strictly a matter of convenience. Since strain is usually the link between the kinematic and the constitutive theories, the convenience of this choice in the context of finite elements is based on two considerations: the ease with which the strain can be computed from the displacements, since the latter are usually the basic variables in the finite element model, and the appropriateness of the strain measure with respect to the particular constitutive model. For example, as mentioned above, it appears that log strain is particularly appropriate to plasticity, while large-strain elasticity analysis (for rubbers and similar materials) can be done quite satisfactorily without ever using any "strain" measure except the stretch ratio ¸. # Strain in general three-dimensional motions Having defined the basic concept of "strain" in one dimension, we now generalize the idea to three dimensions. In \`\`Deformation,'' Section 1.4.1, we established that the deforming part of the motion in the immediate neighborhood of a material point is completely characterized by six variables: the three principal stretch ratios $( \lambda _ { I } , \lambda _ { I I }$ , and $\lambda _ { I I I } )$ and the orientation of the three principal stretch directions in the current (or in the reference) configuration. This immediately gives the generalization of the one-dimensional strain function introduced above. We first choose the function f that will be used as the strain measure. $\varepsilon _ { I } = f { ( \lambda _ { I } ) }$ will be the strain along the first principal direction, $\mathbf { n } _ { I } ; \varepsilon _ { I I } = f ( \lambda _ { I I } )$ will be the strain along $\mathbf { n } _ { I I } ;$ and $\varepsilon _ { I I I } = f ( \lambda _ { I I I } )$ will be the strain along ${ \bf n } _ { I I I }$ . The matrix Equation 1.4.2-2 $$ \pmb {\varepsilon} = \varepsilon_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \varepsilon_ {I I} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \varepsilon_ {I I I} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T} $$ completely characterizes the state of strain at the material point. Notice the resemblance to the definition of the stretch matrix, Equation 1.4.1-10: we might consider " to be defined by the matrix function $$ \boldsymbol {\varepsilon} = f (\mathbf {V}), $$ where we understand a matrix function to mean that the two matrices have the same principal directions with their principal values related by the definition of $f ,$ which is a convenient shorthand way of indicating a relationship between two matrices. In Equation 1.4.2-2 we have written the matrix " by using the principal strain directions in the current configuration. We could equally have begun with the polar decomposition into a stretch followed by rotation of the principal directions of stretch: " would be defined in a similar way and would then be associated with its principal directions in the reference configuration. ABAQUS generally reports strains referred to directions in the current configuration. There is no obvious reason for this choice: either approach would suffice so long as the user knows which is being used. The strain measures reported by ABAQUS are enumerated in \`\`Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. In a finite element code the deformation gradient F is usually computed at each material calculation point from the displacement solution at the nodes of each element and the interpolation function chosen for the element. We now need an algorithm to obtain ", given a choice of strain measure. This algorithm is available immediately from Equation 1.4.1-12: the eigenvalues and eigenvectors of the $3 \times 3$ matrix $\mathbf { F } \cdot \mathbf { F } ^ { T }$ are ${ \lambda _ { I } } ^ { 2 } ; { \lambda _ { I I } } ^ { 2 }$ and ${ \lambda _ { I I I } } ^ { 2 }$ ; and $\mathbf { n } _ { I } , \mathbf { n } _ { I I }$ ; and ${ \bf n } _ { I I I }$ . We can then calculate # Introduction and Basic Equations $\varepsilon _ { I } = f { ( \lambda _ { I } ) }$ , etc. for the function f chosen as the strain measure and, thus, construct $$ \pmb {\varepsilon} = \varepsilon_ {I} \pmb {\mathbf {n}} _ {I} \pmb {\mathbf {n}} _ {I} ^ {T} + \varepsilon_ {I I} \pmb {\mathbf {n}} _ {I I} \pmb {\mathbf {n}} _ {I I} ^ {T} + \varepsilon_ {I I I} \pmb {\mathbf {n}} _ {I I I} \pmb {\mathbf {n}} _ {I I I} ^ {T}. $$ This algorithm also gives principal strain and stretch values--often a useful output because they give a concise description of the state of deformation at a point. However, the algorithm requires computation of the eigenvalues and eigenvectors of a 3 £ 3 matrix at each of many points in the model at each of many iterations, which involves some computational cost. Thus, it would be useful if " could be computed less expensively from F, which is possible only for certain choices of the strain measure, $f ( \lambda )$ . We now consider one such possibility. The unit matrix I can be written as $$ \mathbf {I} = \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}. $$ Using Equation 1.4.1-12, Equation 1.4.2-3 $$ \begin{array}{l} \mathbf {F} \cdot \mathbf {F} ^ {T} - \mathbf {I} = \mathbf {V} \cdot \mathbf {V} - \mathbf {I} \\ = (\lambda_ {I} ^ {2} - 1) \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + (\lambda_ {I I} ^ {2} - 1) \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + (\lambda_ {I I I} ^ {2} - 1) \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}. \\ \end{array} $$ Green's strain was defined in one dimension as $$ \varepsilon^ {G} = \frac {1}{2} (\lambda^ {2} - 1). $$ Comparing this one-dimensional definition with Equation 1.4.2-2 and Equation 1.4.2-3, we see that $$ \pmb {\varepsilon} ^ {g} = \frac {1}{2} (\mathbf {F} \cdot \mathbf {F} ^ {T} - \mathbf {I}) $$ is then a generalization of Green's strain in one dimension. (The more standard definition of Green's strain matrix is obtained by using FT ¢ F instead of F ¢ FT , so the strain matrix is taken on the reference configuration instead of the current configuration as a basis: $$ \pmb {\varepsilon} ^ {G} = \frac {1}{2} (\mathbf {F} ^ {T} \cdot \mathbf {F} - \mathbf {I}). $$ The definition we have adopted is consistent with taking the strain matrix on the current configuration. The only difference between the two definitions is the configuration in which the matrix is defined--whether we think of the motion as rigid body rotation of the principal axes of stretch, R, followed by stretch along those axes, V, or stretch along the principal axes, U, followed by rigid body rotation of those axes, R. The choice is arbitrary.) Green's strain matrix is, thus, available directly from the deformation gradient without first having to solve for the principal directions. This advantage makes Green's strain computationally attractive. Recall that strain is the link between the kinematics and the constitutive theory, so the strain choice should be optimal based on the two considerations of convenience and appropriateness. We have already suggested that logarithmic strain is the most appropriate for elastic-plastic or elastic-viscoplastic materials in which the elastic strains are always small (because the yield stress is small compared to the elastic modulus), so it appears that the computational convenience of Green's strain cannot be used to advantage. However, the choice of a strain function, f(¸), was restricted so that, for small strains but arbitrary rotations, all strain measures are the same to the order of the approximation. Thus, for such cases Green's strain is a very convenient choice for computing the strain. The small-strain, large-rotation approximation is often useful--especially in structural problems (shells and beams) because there the thinness or slenderness of the members often allows large rotations to occur with quite small-strains--and Green's strain is commonly used in large-rotation, small-strain formulations for such problems as shell buckling. Finally, it is worth remarking that the familiar "small-strain" measure used in most elementary elasticity textbooks, $$ \boldsymbol {\varepsilon} = \frac {1}{2} \left(\frac {\partial \mathbf {u}}{\partial \mathbf {X}} + \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] ^ {T}\right), $$ is useful only for small displacement gradients--that is, both the strains and the rotations must be small for this strain measure to be appropriate. This can be demonstrated by considering pure rotation of a specimen: even though the material is not stretched, the components of this measure of strain become nonzero as the rotation increases. # 1.4.3 Rate of deformation and strain increment Many of the materials we need to model are path dependent, so usually the constitutive relationships are defined in rate form, which requires a definition of strain rate. The velocity of a material particle is $$ \mathbf {v} = \frac {\partial \mathbf {x}}{\partial t}, $$ where the partial differentiation with respect to time ( t) means the rate of change of the spatial position, x, of a fixed material particle. Here, again, we take the Lagrangian viewpoint: we observe a material particle and follow it through the motion, rather than looking at a fixed point in space and watching the material flowing through this point. The Lagrangian point of view is used for the mechanical modeling capabilities in ABAQUS because we are usually dealing with history-dependent materials and the Lagrangian perspective makes it easy to record and update the state of a material point since the mesh is glued to the material. The velocity difference between two neighboring particles in the current configuration is $$ d \mathbf {v} = \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \cdot d \mathbf {x} = \mathbf {L} \cdot d \mathbf {x}, $$ where Equation 1.4.3-1 $$ \mathbf {L} = \frac {\partial \mathbf {v}}{\partial \mathbf {x}} $$ is the velocity gradient in the current configuration. In \`\`Deformation,'' Section 1.4.1, we introduced the definition of the deformation gradient matrix, F: $$ d \mathbf {x} = \mathbf {F} \cdot d \mathbf {X}, $$ so $$ d \mathbf {v} = \mathbf {L} \cdot d \mathbf {x} = \mathbf {L} \cdot \mathbf {F} \cdot d \mathbf {X}. $$ We could also obtain the velocity difference directly by $$ d \mathbf {v} = \frac {\partial}{\partial t} (\mathbf {F} \cdot d \mathbf {X}) = \dot {\mathbf {F}} \cdot d \mathbf {X}, $$ where $$ \dot {\mathbf {F}} = \frac {\partial \mathbf {F}}{\partial t}, $$ because dv is defined as the velocity difference between two neighboring material particles and, having chosen these particles, the gauge length between them in the reference configuration, dX, is the same throughout the motion and, so, has no time derivative. Comparing the two expressions for dv in terms of the reference configuration gauge length dX, we see that $$ \mathbf {L} \cdot \mathbf {F} = \dot {\mathbf {F}} $$ or $$ \mathbf {L} = \dot {\mathbf {F}} \cdot \mathbf {F} ^ {- 1}. $$ Now L will be composed of a rate of deformation and a rate of rotation or spin. Since these are rate quantities, the spin can be treated as a vector; thus, we can decompose L into a symmetric strain rate matrix and an antisymmetric rotation rate matrix, just as in small motion theory we decompose the infinitesimal displacement gradient into an infinitesimal strain and an infinitesimal rotation. The symmetric part of the decomposition is the strain rate (it is called the rate of deformation matrix in many textbooks) and is $$ \begin{array}{l} \dot {\pmb {\varepsilon}} = \frac {1}{2} \left(\mathbf {L} + \mathbf {L} ^ {T}\right) \\ = \frac {1}{2} \left(\left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] + \left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] ^ {T}\right). \\ \end{array} $$ The antisymmetric part of the decomposition is the spin matrix, $$ \begin{array}{l} \boldsymbol {\Omega} = \frac {1}{2} \left(\mathbf {L} - \mathbf {L} ^ {T}\right) \\ = \frac {1}{2} \left(\left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] - \left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] ^ {T}\right). \\ \end{array} $$ These are particularly simple and familiar forms; for example, "\_ is identical to the elementary definition of "small strain" if we replace the particle velocity, v, with the displacement, u. In one dimension "\_ is $$ \dot {\varepsilon} = \frac {d v}{d x}, $$ which identifies "\_ as the rate of logarithmic strain, $$ \varepsilon = \ln \lambda . $$ This interpretation would also be correct if the principal directions of strain rotate along with the rigid body motion (because the identification can be applied to each principal value of the logarithmic strain matrix). In the general case, when the principal strain directions rotate independent of the material, "\_ is not integrable into a total strain measure. Nevertheless, the identification of "\_ with the rate of logarithmic strain in the particular case of nonrotating principal directions provides a useful interpretation of the logarithmic measure of strain as a "natural" strain if we think of $\dot { \varepsilon } ,$ as it is defined above as the symmetric part of the velocity gradient with respect to current spatial position, as a "natural" measure of strain rate. The typical inelastic constitutive model requires as input a small but finite strain increment $\Delta \varepsilon ,$ as well as vector and tensor valued state variables (such as the stress) that are written on the current configuration. In ABAQUS/Explicit and for shell and membrane elements in ABAQUS/Standard, a slightly different algorithm is used to calculate $\Delta \mathbf { R }$ . For most element types in ABAQUS/Standard we approach this problem by first using the polar decomposition in the increment to define the change in the average material rotation over the increment, $\Delta \mathbf { R }$ , from the total deformation in the increment, $\Delta \mathbf { F }$ : $$ \Delta \mathbf {F} = \Delta \mathbf {V} \cdot \Delta \mathbf {R}. $$ All vectors and tensors associated with the material (whose values are available at the beginning of the increment from previous calculations) can now be rotated to the configuration at the end of the