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Strain tensorThe strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3dimensional deformation:
The deformation of an object is defined by a tensor field, i.e., this strain tensor is defined for every point of the object. This field is linked to the field of the stress tensor by the generalized Hooke's law. In case of small deformations, the strain tensor is the Green tensor or Cauchy's infinitesimal strain tensor, defined by the equation: Where u represents the displacement field of the object's configuration (i.e., the difference between the object's configuration and its natural state). This is the 'symmetric part' of the Jacobian matrix. The 'antisymmetric part' is called the small rotation tensor. For large (finite) deformations see Finite Deformation Tensors.
Demonstration in simple casesOnedimensional elongationWhen the [AB] segment, parallel to the x_{1}axis, is deformed to become the [A'B' ] segment, the deformation being also parallel to x_{1}
the ε_{11} strain is (expressed in algebraic length): Considering that
the strain is The series expansion of u_{1} is and thus And in general Pure shear strainLet us now consider a pure shear strain. An ABCD square, where [AB] is parallel to x_{1} and [AD] is parallel to x_{2}, is transformed into an AB'C'D' rhombus, symmetric to the first bisecting line.
The tangent of the γ angle is: for small deformations, and and u_{2}(A) = 0. Thus, Considering now the [AD] segment: and thus where γ_{12} is the engineering strain, which is equal to 2γ. It is interesting to use the average because the formula is still valid when the rhombus rotates; in such a case, there are two different angles and and the formula allows for neglecting the variation of angle due to rigidbody motion (which gives no contribution to the strain). Relative variation of the volumeThe dilatation (the relative variation of the volume) δ = ΔV/V_{0}, is the trace of the tensor: Actually, if we consider a cube with an edge length a, it is a quasicube after the deformation (the variations of the angles do not change the volume) with the dimensions and V_{0} = a^{3}, thus as we consider small deformations, therefore the formula.
In case of pure shear, we can see that there is no change of the volume. Derivation of the strain tensorLet the position of a point in a material be specified by a vector with components x_{i}. Let the point then move a small distance to a new position specified by a vector with components where u_{i} is a vector function of . Let x_{i} + dx_{i} be a point close to x_{i}. After the motion, it will be in a new position given by: Since the u_{i} are small, we may approximate them by the first two terms in their Taylor series where we have used to represent and we have used Einstein notation in which repeated indices in a product are assumed to be summed (i.e. index j in this case). is the Jacobian matrix of the u_{i} function. If we represent the unit matrix by δ_{ij} then the above equation may be written: It is seen that the final term (the displacement matrix) specifies the infinitesimal change in the position (dx'_{i}) of the nearby particle. If the u_{i} are constants, the displacement matrix will be the unit matrix, and the resulting displacement will simply be a rigid translation. Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix. Writing the diplacement matrix (in parentheses in the above equation) in this manner yields: The first two terms are the unit matrix and the antisymmetric part of the displacement matrix. These are the first two terms in the Taylor series of a rigid rotation about the translated point x'_{i}. They constitute an infinitesimal rotation and therefore do not represent a deformation of the material. It is the second, symmetric matrix which represents the deformation of the material and this is just the strain tensor : See also
References
Categories: Continuum mechanics  Materials science 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Strain_tensor". A list of authors is available in Wikipedia. 