CN110836825B - Method for in-situ estimation of tensile deformation of rubber-like superelastic material by spherical indentation method - Google Patents
Method for in-situ estimation of tensile deformation of rubber-like superelastic material by spherical indentation method Download PDFInfo
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Abstract
The invention relates to a method for estimating tensile deformation of a rubber-like superelasticity material in situ based on a spherical indentation method, and belongs to the technical field of material analysis and testing. The feasibility of the practical application of the invention mainly depends on the nonlinear mechanical characteristics of the material. After such materials are deformed, the contact area formed by the spherical indentation of the material is in an ellipse, and the eccentricity of the ellipse is changed along with the change of the deformation state. Furthermore, the indentation modulus increases with increasing elongation due to the material's tensile hardening. Therefore, the deformation state of the material can be estimated according to the contact area eccentricity and indentation modulus determined by a spherical indentation experiment. The method is used for estimating the tensile deformation of the rubber super-elastic material, and is the development and the attempt of an indentation method in the practical engineering application.
Description
Technical Field
The invention belongs to the technical field of material analysis and test, and particularly relates to a method for estimating tensile deformation of a rubber-like superelasticity material in situ by using a spherical indentation method.
Background
The indentation method is a simple and easy method for testing the mechanical property of the material, and belongs to a contact type nondestructive testing method. The size of the indentation probe can be as small as micrometer or even nanometer according to the requirement of experimental conditions, such as nano indentation technology. The method calculates the mechanical property of the material according to a load-displacement curve in the indentation process, has simple experimental condition requirements, and is a detection method which is widely applied at present. At present, the application of the indentation method mainly focuses on measuring the mechanical properties of isotropic materials, such as hardness, strength and the like, but the indentation method is weak for more complex materials or structures, such as multilayer materials, composite materials, anisotropic materials and the like. This is mainly due to the mere load-displacement data during indentation testing, and the lack of experimental measurement data limits the application and extension of this method. However, in addition to the load-displacement data, the contact area during indentation is also valuable experimental data that can be obtained by relevant observation means. It is very meaningful to expand the application field of the indentation method and observe a more complex material structure.
Disclosure of Invention
In view of this, the application provides a method for estimating tensile deformation of a rubber-like superelastic material in situ based on a spherical indentation method, and the method is an expansion and innovation of the application field of the indentation method.
In order to achieve the purpose, the invention provides the following technical scheme:
the invention discloses a method for estimating tensile deformation of a rubber-like super-elastic material in situ based on a spherical indentation method, which comprises the following steps of:
s1, taking a material sample in an natural state, and performing indentation test to obtain the indentation modulus M of the material at the moment 0 According to the relation of indentation modulus of isotropic material, mu = M 0 /2(1-v 2 ) (1 + v) obtaining the shear modulus mu of the material, wherein v is the Poisson ratio of the material;
s2, performing spherical indentation test on the point to be tested of the unknown deformation material, and obtaining lambda according to the indentation modulus M and the eccentricity of the contact area at the point 1 And λ 2 Value of (A) 1 And λ 2 The deformation elongation ratios of the material to be detected in two main directions in the indentation contact surface are respectively;
s3, according to the relational expression
And further estimating the engineering stress of different main directions under the deformation, wherein W is a function of the strain energy density of the material.
Further, the application conditions of the estimation method are as follows:
(1) Super elastic materials such as rubber;
(2) The surface of the position to be measured needs to be flat and smooth;
(3) The indentation contact deformation is a small deformation.
Further, in step S2, λ 1 And λ 2 The determination method and the theoretical basis of (1) are as follows:
when a spherical pressure head normally contacts the flat surface of a rubber material, the contact material can be regarded as a semi-infinite body on the premise of small deformation, an arbitrary anisotropic semi-infinite elastic body is imagined, a Cartesian rectangular coordinate system is established, so that an origin of coordinates is located on the boundary of the semi-infinite body, alpha is a unit vector perpendicular to the boundary, and when a unit concentration force perpendicular to the boundary is applied to the origin of coordinates, the displacement of an arbitrary point P on the boundary in the direction i is as follows:
where X represents a position vector of an arbitrary point P on the elastic body in the coordinate system, and ρ = | X |, θ is a position angle of the point P, B -1 Is obtained by inverting a tensor function B (t), wherein the specific expression of the tensor function B (t) can be obtained by the following formula,
in the above formula, t represents an arbitrary unit vector, m 0 、n 0 Representing two unit vectors, t, m 0 、n 0 The three are mutually orthogonal and satisfy the right hand rule of m 0 、n 0 Defining a spatial plane in which the angle is arbitrary
Rotate m simultaneously 0 、n 0 Two new unit vectors m, n are obtained, t, m, n are still orthogonal to each other at this time and follow the right-hand rule, and a symbolic expression (ab) is obtained from any two unit vectors a, b, and the specific expressions of the tensor components are as follows
(ab) ij ＝a i C ijkl b l (3)
Wherein, C ijkl The elastic stiffness matrix of the anisotropic material is expressed, which is clearly defined in generalized Hooke's law, each lower corner mark has a value in the range of 1,2 and 3, and the above formula follows Einstein's convention of summation, and the Cartesian rectangular coordinate system (x, y, z) is set as follows: the origin of coordinates is located on the boundary of the semi-infinite body, the x axis and the y axis are both parallel to the boundary of the semi-infinite body, the z axis is perpendicular to the boundary and outwards, the spherical pressure head acts on the semi-infinite body along the z axis in a displacement loading mode, friction influence is not considered between the pressure head and the semi-infinite body, the problem belongs to the smooth contact problem, the centroid of the elliptical contact area is translated to the origin of coordinates by translating coordinate axes, the initial coordinate system (x, y, z) is rotated to generate a new coordinate system (x ', y ', z '), the rotation angle is beta, the x ' axis and the y ' axis in the new coordinate system are respectively coincident with the symmetrical axis of the contact ellipse, the axial lengths of the elliptical contact area corresponding to the x ' axis and the y ' axis are respectively represented by 2a and 2b, and then the elliptical contact area can be represented as
The effect of a spherical pressure head on a semi-infinite body is understood to be an infinite number of normal concentrated forces acting on the semi-infinite body in an elliptical contact area, assuming a contact stress distribution in the contact area of
Wherein p is 0 The total contact force can be expressed as a parametric quantity, representing the value of the contact stress at the origin, integrating said contact stress over the entire elliptical area
Then there are
Wherein, the first and the second end of the pipe are connected with each other,
the average contact stress within the area of the ellipse is shown,
the elliptic contact surface is divided, and the shadow area infinitesimal can be expressed as
The normal displacement at M point under the action of the infinitesimal area dA can be expressed according to formula (1)
Wherein the content of the first and second substances,
represents the included angle between the straight line MN and the x' axis, s is the distance from the shadow area infinitesimal to the M point, and a function expression
Is for p (x ', y')
And s, using calculus to solve the total normal displacement at boundary point M generated under infinite normal concentrated force on the elliptical contact area into
Wherein the integral formula in formula (9)
The integral interval is a line segment MN, and the space straight line mathematical expression of the MN is
Wherein the content of the first and second substances,
below we define the ellipse as follows
Integral type
The geometric meaning of (A) can be understood as half of the area of the intersecting line ellipse, and the equation x 'of the ellipse is eliminated by the formula (10)' 2 /a 2 +y′ 2 Vb 2 Y' of =1 can be obtained
(b 2 +a 2 k 2 )x′ 2 +2a 3 k 2 x′+a 4 k 2 -a 2 b 2 ＝0 (12)
The two roots of the above formula are coordinate values of the point MN, and the coordinates of the midpoint of the line segment MN can be expressed as
Two axial lengths of the intersecting ellipse (denoted as l) 1 、l 2 ) Are all (x' 0 ，y′ 0 ) Then the axial length can be expressed as
Integral type
Can understand the geometrical meaning ofIs half of the area of the cross line ellipse, then there is the formula of the area of the ellipse
The formula (9) can be simplified into
The expression is the normal displacement of the contact boundary M point when the spherical pressure head acts on the anisotropic semi-infinite body, and the expression of the normal displacement at the N and O points in the same way
Obtaining the following normal displacement expression according to M, N and O points deduced in the foregoing:
H O ＝H M +H N (19)
rotating the coordinate system (x, y, z) by an angle beta (0 beta pi/2) to create a new rectangular coordinate system (x ', y ', z ') such that the contact ellipse is in the new coordinate system, the ellipse symmetry axis coincides with the coordinate axis, and the anisotropic elastomer stiffness matrix C under the original coordinate system ijkl Conversion to C 'in a New coordinate System' ijkl The beta angle is regarded as an unknown parameter, and for any anisotropic semi-infinite body, the displacement of the central point of the contact ellipse formed by the action of the spherical pressure head in the direction of the symmetry axis of the contact area is always zero due to the symmetry, so that the displacement can be written out
Wherein U1 represents the displacement component along the x' axis at the center of the contact ellipse, i.e. the origin, and the second expression in the above expression can be simplified to
Where e = a/b, i.e. the contact ellipse, corresponds to the ratio of the axial lengths of the x 'axis and the y' axis, respectively;
considering the contact geometry, there is a geometric relationship a according to the Pythagorean theorem 2 +(R+H M -H o ) 2 ＝R 2 Thus, the simplification can write
Where R is the radius of curvature of the spherical indenter, it is noted that the geometric relationship described by equation (22) is precise and, similarly, the following can be written
From the aforementioned small deformation assumption, the normal displacement H at the contact edge can be considered M And H N Much smaller than the ram diameter 2R, so that the equations (22), (23) can be simplified to
And
the geometric relation (25) is brought into the formula (16) to obtain
Similarly, the geometric relation (24) is obtained by being brought into the formula (17)
The following relational expressions are not difficult to obtain by combining the expressions (26) and (27)
It can be seen that equation (28) contains only two geometric parameters, namely e and β, so that for any given anisotropic elastic stiffness matrix, two unknown parameters, namely β and e, can be solved and determined by using two nonlinear equation systems (21) and (28). At the same time, the following approximate geometric relation exists
H O ＝abκ (29)
Where K is the indenter curvature, i.e., 1/R,
the coupling (7), (18), (29) makes it possible to obtain the contact force P with respect to the penetration depth H o Approximate analytical expression
The following definition of indentation modulus is given
Where A is the area of the contact ellipse formed by the action of the spherical indenter on the anisotropic semi-infinite body, i.e., π ab, and S is the instantaneous slope of the indentation contact force-contact depth curve.
In the initial state of undeformed material, M 0 =2 μ/(1- ν), it is to be noted that two physical quantities of a and S are available in the ball indentation test, and then the indentation modulus M of the material can be obtained according to equation (31). Next, how to quantitatively describe the instantaneous elastic coefficient of the rubber-like superelastic material in any deformation state, i.e., C in formula (3) ijkl 。
In continuous medium mechanics, the deformation of a superelastic body is usually described by the deformation gradient tensor, i.e. the deformation
Wherein X and X refer to the position vectors of the optional particles of the superelastic continuum in the current configuration and the reference configuration, respectively, and the deformation tensors of the left and right Coxigelin are respectively expressed by a deformation gradient tensor F as B = FF T And C = F T F, the corresponding main invariant of which can be represented by
A neo-Hookean model is selected to describe the strain energy density function of the rubber-type superelasticity material, and the compressibility of the material is considered, and the concrete form is as follows
Wherein, C 10 、D 1 J is the determinant of the deformation gradient tensor F, which is a material parameter,
the first principal invariant of the scaled right Coxigelin deformation tensor is based on a strain energy density function (34), and the initial shear modulus, the volume modulus and the Poisson ratio of an isotropic matrix are respectively
For the convenience of problem description, we set the three main directions of the arbitrary deformation of the superelastic material to be consistent with the coordinate axis directions in the above coordinate system, and the deformation-elongation ratios of the three main directions are λ 1 、λ 2 、λ 3 Then, F, C, B, a are concerned 0 The corresponding concrete expressions are respectively
It is emphasized here that, in the classical theory of linear elastic constitutive
T＝cE,T ij ＝c ijkl E kl (37)
Wherein T is linear stress tensor, E is linear strain tensor, and elastic coefficient c ijkl And first order instant modulus of elasticity component
Are numerically identical, i.e. are
Next, we consider the instant elastic modulus of the material model under any deformation state, and take the x-direction deformation and y-direction deformation elongation ratio as λ 1 、λ 2 Then since the rubber material is almost incompressible, i.e. λ 1 λ 2 λ 3 =1, then λ 3 ＝1/λ 1 λ 2 According to equation (38), the first order instantaneous elastic modulus component of the strain energy density function W at an arbitrary deformation is expressed as follows
Wherein mu is the initial shear modulus of the material and the expression K 0 Mu is related to the Poisson's ratio of the matrix material, where in formula (39) is taken as 1 ＝λ 2 And 1, the elastic coefficient of the material in an initial undeformed configuration is obtained.
It is emphasized that the rubber-like superelastic material belongs to the orthogonal green material, so that β =0.
Substituting formula (39) into formulae (28) and (30) to obtain two compounds containing e, mu, and lambda 1 、λ 2 The equation of (1).
In FIG. 3, the black solid line is the curve of variation of the elliptical eccentricity with the tensile elongation ratio under the condition of pure uniaxial tension in the first main direction of the material; the red dotted line is a curve of the variation of the elliptical eccentricity with the tensile elongation ratio in one direction after the deformation in the second direction is fixed. Although the two curves differ, the eccentricity has a monotonically decreasing trend with a stretch in one direction.
Referring to fig. 4, a black solid line is a curve of variation of the ratio of the indentation modulus to the initial indentation modulus with the tensile elongation ratio under the pure uniaxial tensile condition in the first main direction of the material; the red dotted line is a curve showing the change in the ratio of indentation modulus after fixing the deformation in the second direction to the initial indentation modulus with the tensile elongation ratio in one direction. Similar to the above figure, the ratio of indentation modulus to initial indentation modulus both have monotonically increasing trends as a function of directional stretch, despite the differences between the two curves.
Referring to the above two graphs, it can be concluded that, regardless of the eccentricity of the contact area or the indentation modulus, the change trend is monotonous with the gradual tensile deformation of the rubber-like superelastic material in a certain direction. Thus when the basic mechanical parameters of the material are known; the eccentricity and the indentation modulus of the contact area (two experimental data) obtained according to the indentation experiment can be used for calculating the deformation (lambda) of the material according to the two conditions of the formulas (28) and (30) 1 、λ 2 )。
The invention has the beneficial effects that: the invention discloses a method for estimating tensile deformation of a rubber-like superelasticity material in situ based on a spherical indentation method, according to nonlinear mechanical properties of the rubber-like superelasticity material, because the instant elastic modulus of the material under different deformation configurations is variable and non-isotropic, the contact force is related to the instant elastic modulus of the material when a spherical pressure head is in normal small deformation contact with the flat surface of the rubber material, the geometric shape of the contact area is similar to an ellipse (a circle can be regarded as a special case of an ellipse), and the elliptical eccentricity of the contact area depends on the instant elastic modulus of the material. When the elastic modulus is in time, the elastic modulus presents an isotropic characteristic in a contact plane, and the elliptical contact area tends to be circular; when the contact plane shows the characteristic of anisotropy, the contact area is elliptical, and the more obvious the anisotropy, the smaller the elliptical eccentricity (i.e. the more elliptical), therefore, the indentation method of the invention is different from the conventional indentation method, is specially used for measuring the rubber-like super-elastic material, analyzes the elliptical contact area, and develops and enriches the application of the indentation method.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a meshing of elliptical contact areas;
fig. 2 (a) is cross-sectional contact deformation when y' = 0;
fig. 2 (b) is cross-sectional contact deformation when x' = 0;
FIG. 3 is a graph of elliptical eccentricity versus first principal direction stretch under different constraints;
FIG. 4 is a graph of elliptical indentation modulus as a function of first principal direction stretch under different constraints.
Detailed Description
The following embodiments of the present invention are provided by way of specific examples, and other advantages and effects of the present invention will be readily apparent to those skilled in the art from the disclosure herein. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and embodiments may be combined with each other without conflict.
Wherein the showings are for the purpose of illustrating the invention only and not for the purpose of limiting the same, and in which there is shown by way of illustration only and not in the drawings in which there is no intention to limit the invention thereto; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The same or similar reference numerals in the drawings of the embodiments of the present invention correspond to the same or similar components; in the description of the present invention, it should be understood that if there is an orientation or positional relationship indicated by terms such as "upper", "lower", "left", "right", "front", "rear", etc., based on the orientation or positional relationship shown in the drawings, it is only for convenience of description and simplification of description, but it is not an indication or suggestion that the referred device or element must have a specific orientation, be constructed in a specific orientation, and be operated, and therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes, and are not to be construed as limiting the present invention, and the specific meaning of the terms may be understood by those skilled in the art according to specific situations.
The method for estimating the tensile deformation of the rubber-like super-elastic material in situ based on the spherical indentation method has the following application conditions and ranges,
(1) Super elastic materials such as rubber;
(2) The surface of the position to be measured needs to be flat and smooth;
(3) The indentation contact deformation is a small deformation.
The specific steps are as follows,
s1, taking a material sample in an natural state, and performing indentation test to obtain the indentation modulus M of the material at the moment 0 According to the relation of indentation modulus of isotropic material, mu = M 0 /2(1-v 2 ) (1 + v) to obtain the shear modulus mu of the material, wherein v is the Poisson's ratio of the material. This step is intended to determine the material parameter and may be omitted if the material parameter is known. If the mechanical properties of the material are complex, a method for fitting parameters or other methods for measuring mechanical parameters are required. It should be noted that normally a superelastic material is considered incompressible, but the compressibility of the material is taken into account when choosing a superelastic constitutive model, simply because the matrix of stiffness of the material when incompressible is singular and K is chosen to follow the assumption of incompressible 0 Mu/is set to 10000, which corresponds to a Poisson ratio v of 0.49995, at which point the Poisson ratio of the material is already very close to 0.5, in which case the material is almost similar to an incompressible material.
S2, performing spherical indentation test on the point to be tested of the unknown deformation material, and obtaining lambda according to the indentation modulus M and the eccentricity of the contact area at the point 1 And λ 2 Value of (A) 1 And λ 2 The deformation and elongation ratios of the material to be detected in the first main direction and the second main direction in the indentation contact surface are respectively; according to the formula, the method comprises the following steps of,
the material parameters in the two formulas are determined in S1, and then the initial indentation modulus M 0 Can be determined, and the indentation modulus M and the elliptical eccentricity e under the deformation state to be measured are measured in the indentation experiment in the step, so only lambda is measured 1 And λ 2 Has not been determined. The solution of the two formulas is the solution.
S3, obtaining material reference material deformation by utilizing the previous two steps, and substituting the material into an expression
And further estimating engineering stress in different directions under the deformation, wherein W is a function of the strain energy density of the material.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.
Claims (1)
1. The method for estimating the tensile deformation of the rubber-like superelasticity material in situ based on the spherical indentation method comprises the following application conditions: (1) rubber-like superelastic material; (2) the surface of the part to be measured needs to be flat and smooth; (3) indentation contact deformation is small deformation; the method is characterized by comprising the following steps:
s1, taking a material sample in an natural state, and performing indentation test to obtain the indentation modulus M of the material at the moment 0 According to the relation μ = M of the indentation modulus of isotropic materials 0 /2(1-v 2 ) (1 + v) to obtain the shear modulus mu of the material, wherein v is the Poisson's ratio of the material;
s2, performing spherical indentation test on the point to be tested of the unknown deformation material, and obtaining lambda according to the indentation modulus M and the eccentricity e of the contact area at the point 1 And λ 2 Value of (A) 1 And λ 2 The deformation and elongation ratios of the material to be detected in the first main direction and the second main direction in the indentation contact surface are respectively;
λ 1 and λ 2 The determination method and the theoretical basis of (2) are as follows:
when the spherical pressure head is in normal contact with the flat surface of the rubber material, the contact material can be regarded as a semi-infinite body on the premise of small deformation, an arbitrary anisotropic semi-infinite elastic body is imagined, a Cartesian rectangular coordinate system is established, so that the origin of coordinates is positioned on the boundary of the semi-infinite body,
when a unit concentration force perpendicular to the boundary is applied at the origin, the displacement of any point P on the boundary in the i direction is:
wherein, the first and the second end of the pipe are connected with each other,
represents a position vector of an arbitrary point P on the elastic body in a coordinate system, and
theta is the position angle of the point P,
is a function of tensor
Is obtained by inversion, wherein the tensor function
The specific expression is as follows:
in the above-mentioned formula, the reaction mixture,
which represents an arbitrary unit vector of the vector,
which represents two unit vectors of the vector of the unit,
the three are mutually orthogonal and satisfy the right hand rule of
Defining a spatial plane in which the angle is arbitrary
Rotate at the same time
Two new unit vectors are obtained
At this time
Still remain mutually orthogonal and follow the right hand rule, and furthermore, the symbolic expressions
From arbitrary two unit vectors
The specific expression of the tensor component is obtained as follows
Wherein, C ijkl The elastic stiffness matrix of the anisotropic material is expressed, a definite definition is given in a generalized Hooke law, the value range of each subscript is 1,2 and 3, and the above formula follows Einstein summation convention;
c in formula (3) ijkl The following formula is adopted for representation:
wherein mu is the initial shear modulus of the material and the expression K 0 Mu is related to the poisson's ratio of the matrix material,
while
And
the following relation is satisfied:
equation (6) is the contact force P with respect to the penetration depth H o Approximating an analytical expression;
the following definition of indentation modulus is given:
wherein A is the area of a contact ellipse formed when the spherical pressure head acts on the anisotropic semi-infinite body, S is the instant slope of an indentation contact force-contact depth curve, and A and S can obtain two physical quantities in a spherical indentation test, so that the indentation modulus M of the material can be obtained according to the formula (7);
combining the formulas (1) to (3), substituting the formula (4) into the formulas (5) and (6) to obtain two compounds containing e, mu and lambda 1 、λ 2 Combined with contact area eccentricity e and shear from the ball indentation testShear modulus mu, lambda is obtained according to the formulas (5) and (6) 1 、λ 2 ；
S3, according to the relational expression
And further estimating engineering stress in different main directions under the deformation, wherein W is a function of the strain energy density of the material, and i =1,2,3.
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