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Encyclopedia > Strain tensor

The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ... This article is about the deformation of materials. ... In engineering mechanics, deformation is a change in shape due to an applied force. ...

  • the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ;
  • the other terms εij = 1/2 γij (ij) are the shear strains, i.e. half the variation of the right angle (assuming a small cube of matter before 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 mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...


In case of small deformations, the strain tensor is the Green tensor or Cauchy's infinitesimal strain tensor, defined by the equation:

varepsilon_{ij} = {1 over 2} left ({part u_i over part x_j} + {part u_j over part x_i}right )

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. In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...


For large (finite) deformations see Finite Deformation Tensors. In continuum mechanics, finite deformation tensors are used when the deformation of a body is sufficiently large to invalidate the assumptions inherent in small strain theory. ...

Contents

Demonstration in simple cases

One-dimensional elongation

When the [AB] segment, parallel to the x1-axis, is deformed to become the [A'B' ] segment, the deformation being also parallel to x1


Image:Deformation lineaire et deplacement.png Wikipedia does not have an article with this exact name. ...


the ε11 strain is (expressed in algebraic length):

varepsilon_{11} = frac{Delta l}{l_0} = frac{overline{A'B'}-overline{AB}}{overline{AB}}

Considering that

overline{AA'} = u_1(A) and overline{BB'} = u_1(B)

the strain is

varepsilon_{11} = frac{overline{AB} + overline{BB'}-overline{A'A}}{overline{AB}} - 1
varepsilon_{11} = frac{u_1(B)-u_1(A) + overline{AB}}{overline{AB}} - 1

The series expansion of u1 is As the degree of the taylor series rises, it approaches the correct function. ...

u_1(B) simeq u_1(A) + frac{partial u_1}{partial x_1} cdot overline{AB}

and thus

varepsilon_{11} = frac{partial u_1}{partial x_1}

And in general

varepsilon_{ii} = frac{partial u_i}{partial x_i} = frac{1}{2} left ( frac{partial u_i}{partial x_i} + frac{partial u_i}{partial x_i} right )

Pure shear strain

Let us now consider a pure shear strain. An ABCD square, where [AB] is parallel to x1 and [AD] is parallel to x2, is transformed into a AB'C'D' rhombus, symmetric to the first bisecting line. In physics and mechanics, shear refers to a deformation that causes parallel surfaces to slide past one another (as opposed to compression and tension, which cause parallel surfaces to move towards or away from one another). ... For other uses, see Square. ... For other uses of the word rhombus, see Rhombus (disambiguation) This shape is a rhombus In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i. ...


Image:Cisaillement et deplacement.png Wikipedia does not have an article with this exact name. ...


The tangent of the γ angle is: In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

tan(gamma) = frac{overline{BB'}}{overline{AB}}

for small deformations,

tan(gamma) simeq gamma

and

overline{BB'} = u_2(B) simeq u_2(A) + frac{partial u_2}{partial x_1} cdot overline{AB}

and u2(A) = 0. Thus,

gamma simeq frac{partial u_2}{partial x_1}

Considering now the [AD] segment:

gamma simeq frac{partial u_1}{partial x_2}

and thus

gamma = frac{1}{2} gamma_{12} = varepsilon_{12} = frac{1}{2} left ( frac{partial u_1}{partial x_2} + frac{partial u_2}{partial x_1} right )

where γ12 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 gamma_B = widehat{B'AB} and gamma_D = widehat{D'AD} and the formula allows for neglecting the variation of angle due to rigid-body motion (which gives no contribution to the strain).


Relative variation of the volume

The dilatation (the relative variation of the volume) δ = ΔV/V0, is the trace of the tensor: In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

delta=frac{Delta V}{V_0} = varepsilon_{11} + varepsilon_{22} + varepsilon_{33}

Actually, if we consider a cube with an edge length a, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions a cdot (1 + varepsilon_{11}) times a cdot (1 + varepsilon_{22}) times a cdot (1 + varepsilon_{33}) and V0 = a3, thus

frac{Delta V}{V_0} = frac{left ( 1 + varepsilon_{11} + varepsilon_{22} + varepsilon_{33} + varepsilon_{11} cdot varepsilon_{22} + varepsilon_{11} cdot varepsilon_{33}+ varepsilon_{22} cdot varepsilon_{33} + varepsilon_{11} cdot varepsilon_{22} cdot varepsilon_{33} right ) cdot a^3 - a^3}{a^3}

as we consider small deformations,

1 gg varepsilon_{ii} gg varepsilon_{ii} cdot varepsilon_{jj} gg varepsilon_{11} cdot varepsilon_{22} cdot varepsilon_{33}

therefore the formula.



Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume Wikipedia does not have an article with this exact name. ...


In case of pure shear, we can see that there is no change of the volume.


Derivation of the strain tensor

(Symon (1971) Ch. 10)


Let the position of a point in a material be specified by a vector mathbf{x} with components xi. Let the point then move a small distance to a new position specified by a vector with components

x'_i=x_i+u_i(mathbf{x}),

where ui is a vector function of mathbf{x}. Let xi + dxi be a point close to xi. After the motion, it will be in a new position given by:

x'_i+dx'_i = x_i+dx_i + u_i(mathbf{x}!+!mathbf{dx}),

Since the ui are small, we may approximate them by the first two terms in their Taylor series As the degree of the Taylor series rises, it approaches the correct function. ...

x'_i+dx'_i approx x_i+dx_i + u_i + (partial_j u_i),dx_j

where we have used partial_j to represent partial/partial x_j 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). partial_j u_i is the Jacobian matrix of the ui function. If we represent the unit matrix by δij then the above equation may be written: This article or section does not adequately cite its references or sources. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

x'_i+dx'_i = x'_i + (delta_{ij}+partial_j u_i),dx_j

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 ui 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: In physics, a translation is the operation changing the positions of all objects according to the formula where is a constant vector. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

delta_{ij}+frac{1}{2}(partial_j u_i-partial_i u_j) +frac{1}{2}(partial_j u_i+partial_i u_j)

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 varepsilon_{ij}: A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...

varepsilon_{ij}=frac{1}{2}(partial_j u_i+partial_i u_j)

See also

Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... This article is about the deformation of materials. ... Typical foil strain gauge. ... A stress-strain curve is a graph derived from measuring load (stress - σ) versus extension (strain - ε) for a sample of a material. ... In the uniaxial tensile test commonly carried out to determine some properties of engineering materials, a small testpiece is stretched from an initial, undeformed length to a current, deformed length . ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ...

References

  •  Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7. 

  Results from FactBites:
 
Strain tensor - Wikipedia, the free encyclopedia (615 words)
The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:
(i ≠ j) are the shear strains, i.e.
The deformation of an object is defined by a tensor field, i.e., this strain tensor is defined for every point of the object.
  More results at FactBites »

 
 

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